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ARITHMETIC ALGEBRA GEOMETRY CALCULUS NUMBER THEORY PROBABILITY VECTORS ABSTRACT ALGEBRA

Algebra/Contributing

Notes on contributing to the Calculus textbook. This is the Wikibooks:Local manuals of style for the Calculus textbook.

Note to new contributors

The guidelines given below are intended to keep the organization and style of the book consistent, to reduce the need to rework edits, and to communicate best practices. It is not intended to be a barrier to contributions by would-be contributors. If you have something to contribute but don't have time to read up on all the guidelines given below, go ahead and make your contribution. Other experienced contributors can touch up your edits afterwards so that they conform to the guidelines.

Structure

The structure currently being implemented is as follows:

• The book is divided into chapters and sections, with related sections grouped into supersections (e.g. Integration Techniques within the Integration chapter). These sections are listed on the main page.
• Each section is assigned a section number.
• "==" should be the highest level header used in any section. "=" is reserved for major divisions within the print version of the book.
• Each page including exercises links to a corresponding page of solutions.
• Pages of solutions are organized based on their corresponding page of exercises (exercises set one has a corresponding set one of solutions, etc.).

Conventions

• Use {{Newpage}} templates to separate chapters and sections in the print version.
• Section titles should use title casing while sub-section headers should use sentence casing. Named theorems should be capitalized (e.g. Mean Value Theorem).
• All content pages and section pages should include the {{Calculus/Top Nav}} template at the top and bottom of the page. This template links to the previous and next section or content page. The order of pages defined by the links in the {{Calculus/Top Nav}} template should agree with all tables of contents.
• All content pages and section pages should include {{Calculus/TOC}} just below the bottom {{Calculus/Top Nav}} template.
• Whenever you want to refer to a section by its section number, use {{Calculus/map page|[Section-Name]}}. Then any changes in the ordering of sections can be done easily by changing the {{Calculus/map page}} template.
• Exercises are sequentially numbered within each section.
• Solutions should be complete and logical. Particularly important steps should be noted and described in words as well as being shown symbolically. Naming Convention: Calculus/[Section-Name]/Solutions (ex. Calculus/Differentiation/Solutions)
• Give each figure a figure number. Figure numbering restarts with each new section.
• Use {{nowrap begin}} and {{nowrap end}} for inline math text that shouldn't be broken up across lines.
• Link figure references in the text to the figure's file using {{nowrap begin}}[[:File:Name-of_File|Figure ''k'']]{{nowrap end}}.
• Use "\left(" and "\right)" for parenthesis, "\left|" and "\right|" for absolute values, etc. This makes the LaTeX render better in many cases.
• Use an invisible delimiter (i.e. "\.") with "\|_a^b" for evaluating the antiderivative of an definite integral. The invisible delimiter causes the size of the vertical bar to be sized according to the size of the expression being evaluated.
• Use question-answer templates for exercises like this:

{{question-answer|question=1. Question-goes-here|answer={{noprint|Answer-goes-here}}}}

The noprint template prevents the answer from appearing with the exercise in the print version.

• Solutions (as opposed to answers) go on the solutions page like this:

{{question-answer|question=1. Question-goes-here|answer=Answer-goes-here}}

Bold the answer using either '''Answer''' or <math>\mathbf{Answer}</math> so that people using the print version can quickly find the answer.

• Prefer \frac{a}{b} to {a\over b}. Using consistent markup reduces the need to do multiple searches on different search terms if looking for a particular fraction.
• Use a single space between sentences unless there is a specific reason to do otherwise. Extra whitespace is ignored anyway, so adding extra whitespace only adds to the size of the page without changing its appearance. If you need to add extra whitespace for layout, use "&nbsp;" (the html code for a space) within text or "\,", "\quad", or "\qquad" within <math> tags for small, medium, and large spaces, respectively.

Templates

See Category:Book:User:GoreyCat/Sandbox/Templates for a list of all Calculus textbook templates.

{{Calculus/Top Nav}}

{{Calculus/Top Nav|Limits|Infinite Limits}} produces this navigation box:

All Calculus content pages should include this at the top of the page and at the bottom just above the {{Calculus/TOC}} template.

{{Calculus/TOC}}

{{Calculus/TOC}} produces this navigation box:

All Calculus content pages should include this at the bottom of the page. This also adds the page to the book category.

{{Calculus/map page}}

{{Calculus/map page|Algebra}} returns the section number for the Algebra section, i.e. 1.1. Use this template when you want to refer to a section by its section number. If you re-order the sections, edit the {{Calculus/map page}} template to reflect the new ordering. That way all references in the book will be updated to return the correct section number.

{{Calculus/Def}}

{{Calculus/Def|text=My definition here.}} produces a box for important text:

My definition here.

Use this to introduce significant new definitions and concepts. See Calculus/Limits#Informal definition of a limit, where the informal definition of a limit is inside such a box.

{{Calculus/Stub}}

{{Calculus/Stub}} produces a stub notice, signifying that the given page or section is too short.

On inclusiveness vs. exclusiveness

An Extensions section is included for further topics. Any study beyond fairly basic calculus should be in this section. We should aim to include more than is necessary and ensure that our readers are aware of this. The book should be structured so that J. Random Student can find a rigorous course in the essentials of calculus but also include further study of the topic that readers can pick and choose topics from, as their interests warrant.

TODO

• ADD MORE, ORGANIZE MORE.
• Provide a more in-depth Precalculus section.
• Create problem sets (preferably original, rigorous and realistic).
• Create answer sets.

Pie Chart

Pie charts are best used to compare parts to the whole by percentages. By measuring the number of degrees that a piece of the pie chart is, one can find the percentage it represents.

${\mbox{degrees}}={\mbox{percent}}\cdot {\frac {360}{100}}$

which simplifies to degrees * 18 / 5

Bar Chart

Bar charts are best for plotting the change in something over a period of time. It is nearly the same as a line chart, except that the points are not connected, and instead extend to the bottom of the chart.

Template:User:GoreyCat/Sandbox/Page

Solving linear inequalities involves finding solutions to expressions where the quantities are not equal.

A number on the number line is always greater than any number on its left and smaller than any number on its right. The symbol "<" is used to represent "is less than", and ">" to represent "is greater than".

For example:

<--|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|----->
  -5    -4    -3    -2    -1     0     1     2     3     4     5

From the number line, we can easily tell that 3 is greater than -2, because 3 is on the right side of -2 (or -2 is on the left of 3). We write it as $3>-2$ (or as $-2<3$). We can also derive that any positive number is always greater than negative number.

Consider any two numbers, a and b. One and only one of the following statements can be true:

1. $a>b$,
2. $a=b$, or
3. $a<b$

This is the Law of Trichotomy.

For an inequality with one unknown, there may be many (sometimes infinite) possible solutions.

Properties

• Transitive property:

For any three numbers $x$, $y$, $z$, if $x>y$ and $y>z$, then $x>z$.

• Additive property:

In an inequality, we can add or subtract the same value from both sides, without changing the sign (i.e. ">" or "<"). That is to say, for any three numbers $x$, $y$ and $p$, if $x>y$, then $x+p>y+p$ and $x-p>y-p$.

• Multiplicative property

We can multiply or divide both sides by a positive number without changing the sign. For example, if we have any two numbers $x$ and $y$, and another positive number $p$, then if $x>y$, then $x\cdot p>y\cdot p$ and ${\frac {x}{p}}>{\frac {y}{p}}$.

When we multiply or divide both sides by a negative number, we have to change the sign of the inequality (i.e, ">" change to "<" and vice versa). So if we are given two numbers $x$ and $y$, and another negative number $p$, then if $x>y$, $x\cdot p<y\cdot p$ and ${\frac {x}{p}}<{\frac {y}{p}}$.

Now we can go on to solve any linear inequalities.

Solving Inequalities

Solving inequalities is almost the same as solving linear equations. Let's consider an example: $x+4<13$. All we have to do is to subtract 4 on both sides. We will then get $x<9$, and that is the answer! Note, however, what you get is not a single answer, but a set of solutions, i.e., any number that satisfies the condition $x<9$ (any number that is less than 9) can be a solution to the inequality. It is very convenient to represent the solution using the number line:

<-------------------o
<-+-----+-----+-----+-----+-----+-->
  6     7     8     9     10    11

(Note: the open circle ("o") shows that the value 9 is not included in the solution set, as the inequality of this equation is less than 9, not less than or equal to 9. When we deal with less (greater) than or equal to (≤ or ≥) later on, we use a closed circle ("●") to show that the value is included in the solution set.)

Let us try another more complicated question: $3x-2\geq 2(x-3)$. First, you may want to expand the right hand side: $3x-2\geq 2x-6$. Then we can simply rearrange the terms so that all the unknown variables are on one side of the equation, usually the left hand side: $3x-2x\geq -6+2$. Hence we can easily get the answer: $x\geq -4$. This solution is represented on the number line below. Note that the solution requires a closed circle ("●"), because the $x$ is greater than or equal to 4.

              ●------------------->
<-+-----+-----+-----+-----+-----+-->
 -6    -5    -4    -3    -2    -1

Inequalities with a variable in the denominator

For example consider the inequality

${\frac {2}{x-1}}<2\,$

In this case one cannot multiply the right hand side by $(x-1)$ because the value of x is unknown. Since x may be either positive or negative, you can't know whether to leave the inequality sign as $<$ (ie less than), or reverse it to > (ie greater than). The method for solving this kind of inequality involves four steps:

1. Find out when the denominator is equal to zero. In the above example the denominator equals zero when $x=1$.
2. Pretend the inequality sign is an $=$ sign and solve it as such: ${\frac {2}{x-1}}=2\,$, so $x=2$.
3. Plot the points $x=1$ and $x=2$ on a number line with an unfilled circle because the original equation included < (it would have been a filled circle if the original equation included $\leq$ or $\geq$). You now have three regions: $x<1$, $1<x<2$, and $x>2$.
4. Test each region independently. in this case test if the inequality is true for $1<x<2$ by picking a point in this region (e.g. $x=1.5$) and trying it in the original inequation. For x=1.5 the original inequation doesn't hold. So then try for $1>x>2)$ (e.g. $x=3)$). In this case the original inequation holds, and so the solution for the original inequation is $1>x>2)$.

Compound Inequalities

A compound inequality is a pair of inequalities related by the words and or or. In an and inequality, both inequalities must be satisfied. All possible solution values will be located between two defined numbers, and if this is impossible, the compound inequality simply has no solutions.

Consider this example: $x+6\geq 2$ and $x\leq 2$. First, solve the first inequality for x to get $x\geq -4$. All and inequalities can be rewritten as one inequality, like this: $-4\leq$$x\leq 2$ (write x between two ≤'s or <'s or both with the smaller number on the left and the larger number on the right). Now, we can graph this inequality on a number line as a line segment. Remember, all solutions to ≤ or ≥ must be graphed with closed circles. Interpret this graphic as "all numbers between -4 and 2, including -4 and 2."

        ●-----------------●
<-+-----+-----+-----+-----+-----+-->
 -6    -4    -2     0     2     4

Now, let us consider or inequalities. Or inequalities usually do not have a set of solutions that satisfies both. Instead, they usually have two sets of infinite numbers that are solutions to each one. Because of this, or graphs define which numbers satisfy either equation. For example: $x\;<1$ or $x-1\geq 2$. First, solve for x in the second inequality to get $x\geq 3$. Now, graph the two inequalities on the same number line. Remember to use open and closed circles accordingly.

<-------------o           ●-------->
<-+-----+-----+-----+-----+-----+-->
 -1     0     1     2     3     4

Solving Inequalities with Absolute Value

Since $'''|x|=|-x|'''$ A inequality involving absolute value will have to solved in two parts.

Solving $'''|x-6|<5'''$

The first part would be $'''x-6<5'''$ which gives $x<11$. The second part would be $'''-(x-6)<5'''$ which solved yields $x>1$.

So the answer to $|x-6|<5$ is $1<x<11$

        ●----------------------------><-----------------------------●
<-+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+-->
  0     1     2     3     4     5     6     7     8     9     10    11    12

Graphing Linear Inequalities

The graphing of linear inequalities is very similar to the graphing of linear functions. A linear inequality is written in

Previous: Solving equations
Next: Quadratic functions

The Natural Numbers

Mathematics requires, in order to avoid confusion or absurdity, an unambiguous definition of vocabulary. While this is true of any science, in mathematics this is achieved absolutely through the abstraction of concepts.

However, the full description of math as above requires time, and is nowhere elementary. Also, in order to have a true axiomization, a math built up from the roots, we will have to use the strongest, in the logical sense, statements possible. This makes them powerful for proofs but often sacrifices intuitiveness. Thus, we will concentrate on the later, only rarely using rigour, as necessary.

To do mathematics we must start by finding a way to think about numbers that is as above, unambiguous, while at the same time obvious and natural. We thus come up with the natural numbers.

Intuitively, we define these as the set N = { 1, 2, 3, ... }. We will say a number is a natural number if it belongs to this set. Soon we will find this needs to be expanded even for elementary treatment, but this set alone already has some interesting properties.

If there is a set A such that if we pick an arbitrary number, call it x, and x ∈ N, we say A ⊆ N. In words, A is a subset of N.

By x ∈ A we mean that x is "in" A, or that x is an element of A.

While we have not defined properly a set, membership or inclusion, we have already a feel for what N ought to be like.

Mathematical induction

Principle of mathematical induction

Let A ⊆ N with the following properties:

(i) $1\in A$

(ii) $n\in A$ implies $n+1\in A$

Then A = N

Remark

The principle of mathematical induction is implied by the 'least element principle' (2.1). For, assume that 1.1 holds and let B be the set of all elements of N not in A. If B has any elements (that is, is not empty) then it must have a least one. Call this least element n. Then, by (i), $n\neq 1$. Thus n - 1 is a natural number and is not in B and so is in A. By (ii), $n=(n-1)+1$ is a natural number which is in A. But now n is in both A and B which is impossible. So B must have been empty. That is, A = N.

This serves as a fundamental property of the natural numbers, and we, further defining order, will call these numbers well-ordered, primarily because of this principle. In more rigorous terms, we would have to identify a separate axiom, called the axiom of induction, which encompasses this property. However, for our current purposes, this is sufficient.

Principle of mathematical induction (modified)

Let A be a set of natural numbers which has the following properties:

(i) The natural number $n_{0}$ is in A

(ii) If $n\geq n_{0}$ and $n\in A$, then $n+1\in A$

Then A contains all natural number $n\geq n_{0}$

Numbers

In this section, we shall describe some of the more commonly occurring types of numbers together with some of their properties.

Types of Numbers

Natural Numbers, N

As already mentioned, these are just ordinary counting numbers 1,2,3,4,...,29... . Note that zero has not been included in this set. This varies with different books or mathematicians and may include zero as a natural number.

An important property of the natural numbers is the ordering. Note that natural numbers come with an idea of size so that we can talk about larger and smaller natural numbers.

Integers, Z

By the integers, we mean the natural numbers together with their negatives and zero. Although we all, presumably, have a reasonably clear practical idea of how to work with the integers, there are definite problems as to what they actually are. Referring to such things as the 'number line' does not solve these problems because it relies heavily on our intuition.

One way that mathematicians solve this problem is to make an artificial construction which produces an artificial object with the properties we expect of the integers. Although we do not use this in everyday mathematics, it gives a precise definition which we can use to justify our use of negative numbers and also provides a model we can use when we wish to develop quite new constructions which are not so intuitively reasonable. We shall not describe this for the integers but will give a brief description next of a similar process starting with the integers and producing the rational numbers.

Rational Numbers, Q

With the integers, we have a number system which is closed under addition, multiplication and subtraction. The next step is to produce a collection numbers which is also closed under division (except by zero). This is the rational numbers. Again we should all be able to manipulate rational numbers but there is some problem with what they actually mean. For example, what does it mean to say that ${\frac {2}{3}}={\frac {4}{6}}={\frac {6}{9}}$? We usually want the equals sign to denote the fact that two things are identical and the symbols ${\frac {2}{3}}$ and ${\frac {4}{6}}$ are certainly not identical.

So we redefine what we mean by equality of fractions. Let us consider the set of ordered pairs of integers with the second integer non-zero; that is,

$\mathbb {Q} '=\{(x,y):a,b\in \mathbb {Z}$ and $b\neq 0\}$

(An ordered pair is just a pair in which it is specified which comes first and which second.) We want these ordered pairs to represent rational numbers but on a many-to-one basis. So we redefine equality of these ordered pairs by

$(a,b)\equiv (c,d)$ if and only if ad = bc

(Read this as (a,b) is equivalent to (c,d).) This corresponds to the usual definition of equality of fractions. We can then think of one rational number as being represented by a collection of ordered pairs of integers all equivalent to each other. Two ordered pairs represent the same rational number exactly when they are equivalent. WE can then go on to define, in terms of the ordered pairs, the usual operations of addition, subtraction, multiplication and division. For example,

(a,b) + (c,d) = (ad+bc,bd)

We will define addition of rational numbers by taking an ordered pair for each number and then adding these ordered pairs. There is still a problem however. We need to be sure (without checking each case) that, for example, because $(2,3)\equiv (4,6)\equiv (6,9)$, then also

$(2,3)+(5,6)\equiv (4,6)+(5,6)\equiv (6,9)+(5,6)$

What is order?

Again, as we try to find a way to define, based on the already established ideas of addition, and the intuitive idea of order, in what sense is a number larger than another. In what way can we arrange the numbers?

With N this is easy, we say, if a, b are natural numbers, a < b if there is a natural number c such that a + c = b. From the principle of induction, we can thus order N in the following manner: 1, 2, 3, ... As has been done before with our intuition.

With Z, we run into a problem, where do we start? Since the principle of induction does not apply on Z as a whole, we have to write it as ..., -1, 0, 1, ..., leaving "..." on both sides. However, the definition, as before, applies.

With Q, clearly ${\frac {1}{2}}$ > ${\frac {1}{3}}$, yet ${\frac {1}{6}}$ is not in N and so our definition fails. The new definition would be

Definition: (Order on Q) Let a, b be rational numbers, then a < b if there is a positive rational number c such that a + c = b.

We call a rational c positive if it is equivalent to a rational ${\frac {n}{m}}$ such that n and m are both natural. In other words, if c = ${\frac {p}{q}}$, then c is positive if pq is a natural number. A simple exercise would be to show these are equivalent. (Note: Statements are equivalent if each implies the other.)

Any set such that for each two elements in it we can definitely say a < b, a > b, or a = b is called totally ordered.

The least element property

Any non-empty subset of the natural numbers has a least element. The words 'any non-empty subset' mean that the subset we take should have at least one element in it.

Any set with the least element property is called well-ordered. Thus N is well-ordered as above, while Z and Q are not.

Order properties of natural numbers

(i) There is no largest natural number.

(ii) There are natural numbers which are nearest neighbours in the sense that there is no natural number strictly between them.

Proofs

Definition

A proof is simply a mathematical argument designed to convince the reader of some fact. However an intuitive proof differs from an inductive proof. The former being a fact that can be assumed to be an observable constant from daily experiences (e.g. when two test tubes of milk are poured into one beaker, then if we divide the final volume again into the two beakers, we should get full test tubes again (almost). This explains why ½+½ will still be 1. Since this follows from intuition/observation/experience alone it can be considered an intuitive proof.). The latter, inductive proof, is of the type where many intuitive proofs can be combined to get the desired result. For example, we know that pouring two test tubes of milk gave us two units in the beaker, so if the volume of the test tube can be regarded as a unit volume, then n such test tubes will give us a total of n units of milk in the beaker. Mathematical proof itself is based on natural logic and philosophy. Modern mathematics generally assumes most proofs to be absolute, not without reason though. This allows them to engage in complex problems using purely inductive reasoning. However the basis for all that still remains natural philosophy and logic. This is reflected in Newton's title to a relatively abstract mathematical treatment of his work being titled as 'Principles of natural philosophy'.

Mathematical terms

and - 'A and B' means that both A is true and B is true

not - This has the usual meaning in English; note that not(not(A)) is the same as A

or - This is always what is called 'inclusive or'. That is 'A or B' means that either A is true or B is true or both.

for all - This has the usual meaning in English

there exists - This has the usual meaning in English

implies; if-then - These kinds of statements are at the centre of mathematical reasoning.

if and only if; equivalent - 'A implies B' and 'B implies A'

Theorem, Proposition, Lemma - They all mean roughly the same thing but are in decreasing order of importance.

Corollary - Something which follows easily from a Theorem but for which the statement is not entirely obvious from the statement of the Theorem.

Summation Notation

The summation notation is a convenient abbreviation for sums of several real numbers. If $a_{1}$,$a_{2}$,...,$a_{n}$ are reals, we define

$\sum _{k=1}^{n}a_{k}=a_{1}+a_{2}+...+a_{n}$

The summation index k is often called a dummy index, as it can be replaced by any other letter:

$\sum _{k=1}^{n}a_{k}=\sum _{i=1}^{n}a_{i}=\sum _{j=1}^{n}a_{j}$, etc.

Sometimes it is convenient to start summation from 0 instead of 1, or from some other integral value. For instance,

$\sum _{k=0}^{3}x_{k}=x_{0}+x_{1}+x_{2}+x_{3}$, or $\sum _{j=2}^{5}j^{3}=2^{3}+3^{3}+4^{3}+5^{3}$, etc.

The most important properties of the summation notation can be summarized as

$\sum _{k=1}^{n}(a_{k}+b_{k})=\sum _{k=1}^{n}a_{k}+\sum _{k=1}^{n}b_{k}$ (additive property)

$\sum _{k=1}^{n}(ca_{k})=c\sum _{i=1}^{n}(a_{k})$ (homogeneous property)

$\sum _{k=1}^{n}a_{k}=\sum _{k=0}^{n-1}a_{k+1}=\sum _{k=2}^{n+1}a_{k-1}$ (index translation)

$\sum _{k=1}^{n}(a_{k}-a_{k-1})=a_{n}-a_{0}$ (telescoping)

Functions have an Independent Variable and a Dependent Variable

When we look at a function such as  $f(x)={\frac {1}{2}}x,$  we call the variable that we are changing—in this case  $x\,$ --the independent variable. We assign the value of the function to a variable we call the dependent variable. The reason that we say that  $x\,$  is independent is because we can pick any value for which the function is defined—in this case real  $\mathbb {R}$  is implied—as an input into the function. Once we pick the value of the independent variable the same result will always come out of the function. We say the result is assigned to the dependent variable, since it depends on what value we placed into the function.

Equating  $y\,$  with our function  $y={\frac {1}{2}}x,$  then  $2y=2({\frac {1}{2}}x),$  then  $2y=x,\,$  then  $g(y)=2y.\,$

The independent variable is now  $y\,$  and the dependent variable  $x.\,$

• Note: this is a very unusual case where the ordered pair  $(g(y),y)\,$  is reverse mapped  $y\mapsto x\,$  and corresponding reverses (dependent, independent), (range, domain), and now  $x\,$  must be singular for each and every  $y\,$  corresponding to an horizontal line test of function! It would be less desireable to rotate or swap the positions of axes, the order of coordinate pairs  $(x,y)\,$  and (abscissa, ordinate).

Have we used Algebra to change the nature of the function? Let's look at the results for three functions

$f(x)={\frac {1}{2}}x$	$g(y)=2y\,$	$h(x)=2x\,$
$x\,$   $f(x)\,$       $2\,$     $1\,$     $0\,$     $0\,$   $-2\,$     $-1\,$	$y\,$   $g(y)\,$       $1\,$     $2\,$     $0\,$     $0\,$   $-1\,$     $-2\,$	$x\,$     $h(x)\,$       $1\,$     $2\,$     $0\,$     $0\,$   $-1\,$     $-2\,$

If we look at the table above we can see that the independent variable for  $f(x)\,$  gives the same results as the dependent variable of  $g(y).\,$  We can see what this means when we look at the values for  $h(x).\,$  The function  $g(y)\,$  is the same as the function  $f(x),\,$  but when we switch which variable we use as the independent variable between  $g(y)\,$  and  $h(x)\,$  we see that we have discovered that  $g(y)\,$  and  $h(x)\,$  are inverse functions.

Let's take a look at how we can draw functions in  $x\,$  and  $y\,$  and then come back and look at this idea of independent and dependent variables again.

Explicit and Implicit Functions

Variables like  $x\,$  and  $y\,$  formulate a 'relation' using simple algebra.  $f,\,$   $g,\,$  and   $h\,$  commonly denote functions. Function notation  $f(x),\,$  read "eff of ex", denotes a function with 'explicit' dependence on the independent variable  $x.\,$  By assigning variable  $y\,$  to  $f(x),\,$   $y=f(x),\,$   $y\,$  is now an 'implicit' function of   $x\,$  using equation notation. If  $f(x)\,$  is  ${\frac {x}{2}},$  then  $y={\frac {x}{2}}$   [ $y(x)\,$  would denote an 'explicit' function of  $x\,$ ]. A relation is also a function when the dependent variable has one and only one value for each and every independent variable value.

The Cartesian Coordinate System

The Cartesian Coordinate System is a uniform rectangular grid used for plane graph plots. It's named after pioneer of analytic geometry, 17th century French mathematician René Descartes, whose Latinized name was Renatus Cartesius. Recall that each point has a unique location, different from every other point. We know that a line is a collection of points. If we pick a direction of travel for the line that starts at a point then all of the other points can be thought of as either behind our starting point or ahead of it. Finally, a plane can be thought of as a collection of lines that are parallel to each other. We can draw another line that is composed of one point from each of the lines that we chose to fill our plane. If we do this then we can locate the other lines as behind or ahead of the line with the point we chose to start on. Descartes decided to pick a line and call it the  $x\,$ -axis, and to then pick a line perpendicular to this line and call it the  $y\,$ -axis. He then labeled this intersection point  $(0,0)\,$  and origin O. The points to the left (or behind) of this point each represent a negative number that we label as  $(-x,0).\,$  The points to the right (or ahead) of this point each represent a positive number that we label as  $(x,0).\,$  The points on the  $y\,$ -axis that are above  $0,0\,$  are labeled as positive  $(0,y),\,$  and the points on the  $y\,$ -axis below  $0,0\,$  are labeled as negative  $(0,-y).\,$  A point is plotted as a location on the plane using its coordinates from the grid formed by the  $x\,$  and  $y\,$ -axes. If you draw a line perpendicular to the  $x\,$ -axis from a point you pick then that point has the same  $x\,$ -coordinate as the point where that line crosses the  $x\,$ -axis. If you draw a line perpendicular to the  $y\,$ -axis from your point then it has the same  $y\,$ -coordinate as the point where that line crosses the  $y\,$ -axis. If you need to sharpen your knowledge in this area, this link/section should help: The Coordinate (Cartesian) Plane

An equation and its graph can be referred to as equal. This is true since a graph is a representation of a specific equation. This is because an equation is a group of one or more variables along with one or more numbers and an equal sign (  $x=1,\,$  $y=x+1,\,$  and  $y=x^{2}+2x+1\,$  are all examples of equations). Since variables were introduced as way of representing the many possible numbers that could be plugged into the equation. A graph of an equation is a way of drawing the relationship between the numbers that can be input (the independent variable) and the possible outputs that would be produced. For example, in the equation:  $y=x+2,\,$  we could choose to make the  $x\,$  the independent variable and the output number would be two more than the input number every time. The graph of this equation would be a picture showing this relationship. On the graph, each  $y\,$ -value (the vertical axis) would be two higher than the (horizontal)  $x\,$ -value that is plugged in because of the  $+\,2\,$  in the equation.

Linear Equations and Functions

This section shows the different ways we can algebraically write a linear function. We will spend some time looking at a way called the "slope intercept form" that has the equation  $y=f(x)=mx+b\,.\,$

• Linear Equations and Functions

Unless a domain for  $x\,$  is otherwise stated, the domain for linear functions will be assumed to be all real numbers  $\mathbb {R}$  and so the lines in graphs of all linear functions extend infinitely in both directions. Also in linear functions with all real number domains, the range of a linear function may cover the entire set of real numbers for  $y\,,\,$  one exception is when the slope  $m=0\,$  and the function equals a constant. In such cases, the range is simply the constant. Another would be a squaring function where the range would be non-negative when  $b=0\,.\,$

The y-intercept constant b

It was shown that  $y=f(x)=mx+b\,$  has infinite solutions (in the UK,  $y=mx+c\,;\,$  also common  $y=ax+b\,,\,$  $y=a_{1}x+a_{0}\,$  and  $y=-{\frac {A}{B}}x-{\frac {C}{B}}\,$ ). Points  $(x,y)\,$  will be mapped with independent variable  $x\,$  assuming the horizontal axis and  $y\,$  vertical on a Cartesian grid. By assigning  $x\,$  to a value and evaluating  $y\,,\,$  a (single) point coordinate solution is found. When  $x=0\,,\,$  then by zero-product property term  $m\times x=0\,$ ,  and by additive identity terms  $0+b=b=y\,.\,$  The point  $(0,b)\,$  is the unique member of the line (linear equation's solution) where the y-axis is 'intercepted'. More about intercepts link:  The  $x\,$  and  $y\,$  Intercepts

What does the m tell us when we have the equation $y=f(x)=mx+b\,$ ?

$m\,$  is a constant called the slope of the line. Slope indicates the steepness of the line.

Slope

Two separate points fixed anywhere defines a unique straight line containing the points. Confining this study to plane geometry ($R^{2}$) and fixing coordinates for unique points at  $(x_{1},y_{1})\,$  and  $(x_{2},y_{2})\,$  a straight line is defined relating two variables in a linear-equation mappable on a graph-plot. When the two points are identical, infinite lines result, even in a single plane. When  $x_{1}=x_{2}\,$  then a vertical-line mere relation is defined, not a function. Functions are equation-relations evaluating to singularly unique dependent values. Only when (iff)  $x_{1}\neq x_{2},\,$  then is the line containing the points a linear 'function' of  $x.\,$

For a linear function, the slope can be determined from any two known points of the line. The slope corresponds to an increment or change in the vertical direction divided by a corresponding increment or change in the horizontal direction between any different points of the straight line.
Let $\Delta y=\,$ increment or change in the $y\,$ -direction (vertical) and
Let $\Delta x=\,$ increment or change in the $x\,$ -direction (horizontal).
For two points  $(x_{1},y_{1})\,$  and  $(x_{2},y_{2}),\,$  the slope of the function line m is given by: $m={\frac {\Delta y}{\Delta x}}={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$

• This formula is called the formula for slope measure but is sometimes referred to as the slope formula.

For a linear function, fixing two unique points of the line or fixing the slope and any one point of the line is enough to determine the line and identify it by an equation. There is an equation form for a linear function called the point-slope form of a line² which uses the slope  $m\,$  and any one point  $(x_{1},y_{1})\,$  to determine a valid equation for the function's line: $y-y_{1}=m(x-x_{1})\,$
Algebra/Slope

The Pythagorean Theorem and The Distance Formula

Other forms of linear equations?

Algebra/Standard Form and Solving Slope

Intercept Form of a Line

There is one more general form of a linear function we will cover. This is the intercept form of a line, where the constants a and b are such that (a,0) is the x-intercept point and (0,b) is the
y-intercept point.

${\frac {x}{a}}+{\frac {y}{b}}=1\qquad$    where a ≠ 0 and b ≠ 0

Neither constant a nor b can equal 0 because division by 0 is not allowed. The intercept form of a line cannot be applied when the linear function has the simplified form y = m x because the
y-intercept ordinate cannot equal 0.

Multiplying the intercept form of a line by the constants a and b will give

$bx+ay=ab\$

which then becomes equivalent to the general linear equation form A x + B y + C where A = b, B = a, and C = ab. We now see that neither A nor B can be 0, therefore the intercept form cannot represent horizontal or vertical lines. Multiplying the intercept form of a line by just b gives

${\frac {b}{a}}x+y=b$

if we subtract

${\frac {b}{a}}x$

we get:

$y=b-{\frac {b}{a}}x$

which can, in turn, be rearranged to:

$y=-{\frac {b}{a}}x+b$

which becomes equivalent to the slope-intercept form where the slope m = -b/a.

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Example: A graphed line crosses the x-axis at -3 and crosses the y-axis at -6. What equation can represent this line? What is the slope?

Solution: intercept form: $\qquad {\frac {x}{-3}}+{\frac {y}{-6}}=1$

Multiplying by -6 gives ${\frac {-6}{-3}}x+y=-6$

$y=-2x-6\$

so we see the slope m = -2.

     Graph of y = - 2x – 6 showing intercepts.

The line can also be written as $-6x-3y=(-3)(-6)\$

$6x+3y=-18\$

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Example: Can the equation

${\frac {x}{2}}+{\frac {y}{4}}=0$

be transformed into an intercept form of a line, (x/a) + (y/b) =1, to find the intercepts?

Solution: No, no amount of valid mathematical manipulation can transform it into the intercept form. Instead multiplying by 4, then subtracting 2x gives

$y=-2x\$

which is of the form y = m x where m = -2. The line intersects the axes at (0,0). Since the intercepts are both 0, the general intercept form of a line cannot be used.

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Example: Find the slope and function of the line connecting the points (2,1) and (4,4).

Solution: When calculating the slope of a straight line from two points with the preceding formula, it does not matter which is point 1 and which is point 2. Let's set (x₁,y₁) as (2,1) and (x₂,y₂) as (4,4). Then using the two-point formula for the slope m:

$m={\frac {4-1}{4-2}}={\frac {3}{2}}$

Using the point-slope form:

One substitutes the coordinates for either point into the point-slope form as x₁ and y₁. For simplicity, we will use x₁=2 and y₁=1.

$y-1={\frac {3}{2}}(x-2)$

$y-1={\frac {3}{2}}x-3\,$

$\ y={\frac {3}{2}}x-2$

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Using the slope-intercept form:

Alternatively, one can solve for b, the y-intercept ordinate, in the general form of a linear function of one variable, y = m x + b.

$b=y-mx\$

Knowing the slope m, take any known point on the line and substitute the point coordinates and m into this form of a linear function and calculate b. In this example, (x₁,y₁) is used.

$b=y_{1}-mx_{1}\,$

$b=1-{\frac {3}{2}}\cdot 2=1-3=-2$

Now the constants m and b are both known and the function is written as

$y={\frac {3}{2}}x-2$      or alternatively as      $f(x)={\frac {3}{2}}x-2$

__________end of example__________

For another explanation of slope look here:

Slope

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Example: Graph the equation 5x + 2y = 10 and calculate the slope.

Solution: This fits the general form of a linear equation, so finding two different points are enough to determine the line. To find the x-intercept, set y = 0 and solve for x.

$5x+2\cdot 0=10=5x\$

$x=10/5=2\$

so the x-intercept point is (2,0). To find the y-intercept, set x = 0 and solve for y.

$5\cdot 0+2y=10=2y\$

$y=10/2=5\$

so the y-intercept point is (0,5). Drawing a line through (2,0) and (0,5) would produce the following graph.

Graph of 5x + 2y = 10 showing intercepts

To determine the slope m from the two points, one can set (x₁,y₁) as (2,0) and (x₂,y₂) as (0,5), or vice versa and calculate as follows:

$m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}={\frac {5-0}{0-2}}=-{\frac {5}{2}}=-2.5$

__________end of example__________

Summary of General Equation Forms of a Line

The most general form applicable to all lines on a two-dimensional Cartesian graph is

$Ax+By=C\$

with three constants, A, B, and C. These constants are not unique to the line because multiplying the whole equation by a constant factor gives a new set of valid constants for the same line. When B = 0, the rest of the equation represents a vertical line, which is not a function. If B ≠ 0, then the line is a function. Such a linear function can be represented by the slope-intercept form which has two constants.

slope-intercept form:

$y=mx+b\$

The two constants, m and b, used together are unique to the line. In other words, a certain line can have only one pair of values for m and b in this form.

The point-slope form given here

$y-y_{1}=m(x-x_{1})\,$

uses three constants; m is unique for a given line; x₁ and y₁ are not unique and can be from any point on the line. The point-slope cannot represent a vertical line.

The intercept form of a line, given here,

${\frac {x}{a}}+{\frac {y}{b}}=1$    a ≠ 0 and b ≠ 0

uses two unique constants which are the x and y intercepts, but cannot be made to represent horizontal or vertical lines or lines crossing through (0,0). It is the least applicable of the general forms in this summary.

Of the last three general forms of a linear function, the slope-intercept form is the most useful because it uses only constants unique to a given line and can represent any linear function. All of the problems in this book and in mathematics in general can be solved without using the point-slope form or the intercept form unless they are specifically called for in a problem. Generally, problems involving linear functions can be solved using the slope-intercept form
(y = m x + b) and the formula for slope.

Discontinuity in Otherwise Linear Equations

Let variable y be dependent upon a function of independent variable x

$y=f(x)\,,$  also  $y(x)=f\,.$

y is also the function f, and x is also the argument ( ). Let y be the expressed quotient function

$y={\frac {2x^{2}-5x+3}{x-1}}.$

The graph of y's solution plots a continuous straight line set of points except for the point where x would be 1. Evaluation of the denominator with $x=1$ results in division by zero, an undefined condition not a member element of R and outside algebraic closure. y has a discontinuity (break) and no solution at point 1,-1. It becomes important to treat each side of a break separately in advanced studies.

y's otherwise linear form can be expressed by an equation removed of its discontinuity. Factor $x-1$ from the $2x^{2}-5x+3$ numerator (use synthetic division).

$y={\frac {2x^{2}-5x+3}{x-1}}.$

$y={\frac {(2x-3)(x-1)}{(x-1)}}.$

$y=2x-3\,,\;x\neq 1$  (for all x except 1) .

Reducing its (x-1) multiplicative inverse factors (reciprocals) to multiplicative identity (unity) leaves the $(2x-3)$ factor (with implied universal-factor 1/1). Limiting this simpler function's domain; 'all $x$ except $x=1$, where x is undefined' or simply 'and x ≠ 1' (implying 'and R² '); equates it to the original function. This expression is a linear function of x, with slope m = 2 and a y-intercept ordinate of -3. The expression $2x-3$ evaluates to -1 at x = 1, but function y is undefined (division by zero) at that point. There is a discontinuity for function y at x = 1. Practically the function has a sort of one-point hole (a skip), shown on the graph as a small hollow circle around that point. Lines, rays and line segments (and arcs, chords and curves) are shown discontinuous by dashed or dotted lines.

Note: non-linear equations may also be discontinuous—see the subsequent graph plot of the reciprocal function y = 1/x, in which y is discontinuous at x = 0 not just for a point, but over a 'double' asymptotic extremum pole along the y-axis. As x is evaluated at smaller magnitudes (both – and +) closer to zero, y approaches no definition in both the – and + mappings of the function.

Example: What would the graph of the following function look like?

$y={\frac {x^{2}-4}{x+2}}.$

Solution:

$y={\frac {x^{2}-4}{x+2}}={\frac {(x+2)(x-2)}{(x+2)}}.$

Reduce the reciprocal (x + 2) factors to unity. This makes y = x – 2 for all x except x = -2, where there is a discontinuity. The line y = x – 2 would have a slope m = 1 and a
y-intercept ordinate of -2. So for the final answer , we graph a line with a slope of 1 and a y-intercept of -2, and we show a discontinuity at x = -2, where y would otherwise have been equal to -4.

Example: Write a function which would be graphed as a line the same as y = 2 x – 3 except with two discontinuities, one at x = 0 and another at x = 1.

Solution: The function must have a denominator with the factors

denominator = (x – 0)(x – 1) = x (x – 1) .

to have 'zeros' at the two x values. The function's numerator also gets the factors preserving an overall factor of unity, the expressions are multiplied out:

$y={\frac {(2x-3)x(x-1)}{x(x-1)}}={\frac {2x^{3}-5x^{2}+3x}{x^{2}-x}}.$

__________end of example__________