7.
![{\displaystyle \{x:3>3x\}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0538ba1d6d510b33aa633c3fa7ac66782385eacd)
8.
![{\displaystyle \{x:0\leq 2x+1<3\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ddd7a7e1ccc36e286ca07b88c1e51e7ca4f347)
9.
![{\displaystyle \{x:5<x{\mbox{ and }}x<6\}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0fc399ad393851482d35744116cf38968f3b306)
10.
![{\displaystyle \{x:5<x{\mbox{ or }}x<6\}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56d49efdd79ea60fc3157e1d7e0e33aa2c2efc63)
It helps to draw a picture to determine the set of numbers described:
![](//upload.wikimedia.org/wikipedia/commons/0/07/5_lt_x_or_x_lt_6.png)
A number in the set can be on either the red or blue line, so the entire number line is included.
![{\displaystyle \mathbf {(-\infty ,\infty )} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e59432a411c56552e1a2e99e4ae1235d63fdb0a)
It helps to draw a picture to determine the set of numbers described:
![](//upload.wikimedia.org/wikipedia/commons/0/07/5_lt_x_or_x_lt_6.png)
A number in the set can be on either the red or blue line, so the entire number line is included.
![{\displaystyle \mathbf {(-\infty ,\infty )} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e59432a411c56552e1a2e99e4ae1235d63fdb0a)
17.
![{\displaystyle |x+y|=|x|+|y|\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/894551b1785e3a4d1198941463e32bf33d0a50a2)
Let
![{\displaystyle x=-5,y=5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f7f5d0cc2793dd3142d49fb4b10807a20d8d11a)
. Then
, and
![{\displaystyle |x|+|y|=|-5|+|5|=5+5=10}](https://wikimedia.org/api/rest_v1/media/math/render/svg/134e95d8733f3ba1992dda79fd8c27dc8b88612a)
Thus, ![{\displaystyle |x+y|\neq |x|+|y|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddde86bd7b7f2d5f8189c72e9a3d57fdc68cdac8)
falseLet
![{\displaystyle x=-5,y=5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f7f5d0cc2793dd3142d49fb4b10807a20d8d11a)
. Then
, and
![{\displaystyle |x|+|y|=|-5|+|5|=5+5=10}](https://wikimedia.org/api/rest_v1/media/math/render/svg/134e95d8733f3ba1992dda79fd8c27dc8b88612a)
Thus, ![{\displaystyle |x+y|\neq |x|+|y|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddde86bd7b7f2d5f8189c72e9a3d57fdc68cdac8)
false
18.
![{\displaystyle |x+y|\geq |x|+|y|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5139cd95cc368215289201617a4d54aa16fdff8e)
Using the same example as above, we have
![{\displaystyle |x+y|\ngeq |x|+|y|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdcc7483718333bdb1b2df309c2c8287c4880fd3)
.
falseUsing the same example as above, we have
![{\displaystyle |x+y|\ngeq |x|+|y|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdcc7483718333bdb1b2df309c2c8287c4880fd3)
.
false
19.
![{\displaystyle |x+y|\leq |x|+|y|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b02fe24bab89ece41c4640d04bb5e8996ef92ee)
52. Let
.
a. Compute
![{\displaystyle f(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc54adf96acf7e2466de00b7739144d36840108f)
and
![{\displaystyle f(2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65f247dc5542263d64ef5f4346fc7cae009740f4)
.
b. What are the domain and range of
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
?
The domain is
![{\displaystyle (-\infty ,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8c11c44279888c9e395eeb5f45d121348ae10a)
; the range is
![{\displaystyle [0,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc2d914c2df66bc0f7893bfb8da36766650fe47)
,
The domain is
![{\displaystyle (-\infty ,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8c11c44279888c9e395eeb5f45d121348ae10a)
; the range is
![{\displaystyle [0,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc2d914c2df66bc0f7893bfb8da36766650fe47)
,
c. Does
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
have an inverse? If so, find a formula for it.
No, since
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
isn't one-to-one; for example,
![{\displaystyle f(-1)=f(1)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bfef0a3926ccd01d31b2a27bf10ec134a0e493a)
.
No, since
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
isn't one-to-one; for example,
![{\displaystyle f(-1)=f(1)=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bfef0a3926ccd01d31b2a27bf10ec134a0e493a)
.
53. Let
,
.
- a. Give formulae for
i.
![{\displaystyle f+g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d94a24abd865f6f9fd67a7df7e531cae1c769b3)
![{\displaystyle (f+g)(x)=x+2+1/x=(x^{2}+2x+1)/x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d96ccacd274aa20f86e1e520821ae5cf8c3de74)
.
![{\displaystyle (f+g)(x)=x+2+1/x=(x^{2}+2x+1)/x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d96ccacd274aa20f86e1e520821ae5cf8c3de74)
.
ii.
![{\displaystyle f-g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3f3019b383024c33e03b71a287d195f958ca89f)
![{\displaystyle (f-g)(x)=x+2-1/x=(x^{2}+2x-1)/x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e251573db5b0ddd32b18f3305b426c5b01150f7)
.
![{\displaystyle (f-g)(x)=x+2-1/x=(x^{2}+2x-1)/x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e251573db5b0ddd32b18f3305b426c5b01150f7)
.
iii.
![{\displaystyle g-f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfadf2c2279e1c4e1ad815ca1346fe276cfeef64)
![{\displaystyle (g-f)(x)=1/x-x-2=(1-x^{2}-2x)/x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0508b33cdea3fabe17666de5a4532e0d4aa2d32d)
.
![{\displaystyle (g-f)(x)=1/x-x-2=(1-x^{2}-2x)/x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0508b33cdea3fabe17666de5a4532e0d4aa2d32d)
.
iv.
![{\displaystyle f\times g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/173e64c38905633d16e671ea4436592d280d5c6a)
![{\displaystyle (f\times g)(x)=(x+2)/x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ebde5f493aa8a8e6daaf26bb6ede98434e34956)
.
![{\displaystyle (f\times g)(x)=(x+2)/x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ebde5f493aa8a8e6daaf26bb6ede98434e34956)
.
v.
![{\displaystyle f/g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c6b7962d3532e248f07cd42b1bdc9e007b137d)
![{\displaystyle (f/g)(x)=x(x+2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35b99643e85a895311e82ff915564f00d9de3b7b)
provided
![{\displaystyle x\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5)
. Note that 0 is not in the domain of
![{\displaystyle f/g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c6b7962d3532e248f07cd42b1bdc9e007b137d)
, since it's not in the domain of
![{\displaystyle g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
, and you can't divide by something that doesn't exist!
![{\displaystyle (f/g)(x)=x(x+2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35b99643e85a895311e82ff915564f00d9de3b7b)
provided
![{\displaystyle x\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5)
. Note that 0 is not in the domain of
![{\displaystyle f/g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c6b7962d3532e248f07cd42b1bdc9e007b137d)
, since it's not in the domain of
![{\displaystyle g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
, and you can't divide by something that doesn't exist!
vi.
![{\displaystyle g/f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b5280b920015c4f40676a94a1451e90aab3b066)
![{\displaystyle (g/f)(x)=1/[x(x+2)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9420d62f75928eb145c0933455cda94fe9b406be)
. Although 0 is still not in the domain, we don't need to state it now, since 0 isn't in the domain of the expression
![{\displaystyle 1/[x(x+2)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c88b915e8cb5ba979495a707385342f8f93b6274)
either.
![{\displaystyle (g/f)(x)=1/[x(x+2)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9420d62f75928eb145c0933455cda94fe9b406be)
. Although 0 is still not in the domain, we don't need to state it now, since 0 isn't in the domain of the expression
![{\displaystyle 1/[x(x+2)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c88b915e8cb5ba979495a707385342f8f93b6274)
either.
vii.
![{\displaystyle f\circ g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae)
![{\displaystyle (f\circ g)(x)=1/x+2=(2x+1)/x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fe855a026689ee3e1e5c4927f5154de148de2f0)
.
![{\displaystyle (f\circ g)(x)=1/x+2=(2x+1)/x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fe855a026689ee3e1e5c4927f5154de148de2f0)
.
viii.
![{\displaystyle g\circ f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92)
![{\displaystyle (g\circ f)(x)=1/(x+2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a09a232048e3af45c3f0ffdd749c9bddd4f90e)
.
![{\displaystyle (g\circ f)(x)=1/(x+2)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2a09a232048e3af45c3f0ffdd749c9bddd4f90e)
.
b. Compute
![{\displaystyle f(g(2))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68b182b2d5281d9e15e19113cf8245699589a88a)
and
![{\displaystyle g(f(2))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e16ca437e34dfba2b9b6d03273997950bcb2ba51)
.
![{\displaystyle f(g(2))=5/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1cbefefc876bc8939803b88daf0cf28f88c20f)
;
![{\displaystyle g(f(2))=1/4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81981c077c537a0467e030b0e32daa0a967afe65)
.
![{\displaystyle f(g(2))=5/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1cbefefc876bc8939803b88daf0cf28f88c20f)
;
![{\displaystyle g(f(2))=1/4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81981c077c537a0467e030b0e32daa0a967afe65)
.
c. Do
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
and
![{\displaystyle g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
have inverses? If so, find formulae for them.
Yes;
![{\displaystyle f^{-1}(x)=x-2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2df6469872655bb93fc5c36c9659d2acd83a8a9)
and
![{\displaystyle g^{-1}(x)=1/x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4560ebbc9e19c9731a0f30953bea0b714e3031b8)
. Note that
![{\displaystyle g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
and its inverse are the same.
Yes;
![{\displaystyle f^{-1}(x)=x-2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2df6469872655bb93fc5c36c9659d2acd83a8a9)
and
![{\displaystyle g^{-1}(x)=1/x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4560ebbc9e19c9731a0f30953bea0b714e3031b8)
. Note that
![{\displaystyle g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77)
and its inverse are the same.
54. Does this graph represent a function?
![](//upload.wikimedia.org/wikipedia/commons/thumb/a/ac/Sinx_over_x.svg/180px-Sinx_over_x.svg.png)
As pictured, by the Vertical Line test, this graph represents a function.
As pictured, by the Vertical Line test, this graph represents a function.
55. Consider the following function
![{\displaystyle f(x)={\begin{cases}-{\frac {1}{9}}&{\mbox{if }}x<-1\\2&{\mbox{if }}-1\leq x\leq 0\\x+3&{\mbox{if }}x>0.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07cfb6bfa8c4d469b5d3de28d09b7137aab50df8)
a. What is the domain?
![{\displaystyle {(-\infty ,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbe6022b6573765e468386496b2a13afc86ac14)
![{\displaystyle {(-\infty ,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbe6022b6573765e468386496b2a13afc86ac14)
b. What is the range?
![{\displaystyle {(-1/9,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2459612dcaab03af308902d8224629742854b8c)
![{\displaystyle {(-1/9,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2459612dcaab03af308902d8224629742854b8c)
c. Where is
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
continuous?
![{\displaystyle {x>0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac736ec20e9d9cc9ecd79c28ab1a026009a63972)
![{\displaystyle {x>0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac736ec20e9d9cc9ecd79c28ab1a026009a63972)
56. Consider the following function
![{\displaystyle f(x)={\begin{cases}x^{2}&{\mbox{if }}x>0\\-1&{\mbox{if }}x\leq 0.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/804d6e22d89ac6f98dcbec04f06d366a0854bb5e)
a. What is the domain?
![{\displaystyle {(-\infty ,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbe6022b6573765e468386496b2a13afc86ac14)
![{\displaystyle {(-\infty ,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbe6022b6573765e468386496b2a13afc86ac14)
b. What is the range?
![{\displaystyle {(-1,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/973e3ae5258477828503e4729eefb3ede55cff6f)
![{\displaystyle {(-1,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/973e3ae5258477828503e4729eefb3ede55cff6f)
c. Where is
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
continuous?
![{\displaystyle {x>0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac736ec20e9d9cc9ecd79c28ab1a026009a63972)
![{\displaystyle {x>0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac736ec20e9d9cc9ecd79c28ab1a026009a63972)
57. Consider the following function
![{\displaystyle f(x)={\frac {\sqrt {2x-3}}{x-10}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f664cef36f14a26c2836def898294ce65e540e6)
a. What is the domain?
![{\displaystyle {(3/2,10)\cup (10,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/615896bd67b960440e6761d8b55c4c799760cad9)
![{\displaystyle {(3/2,10)\cup (10,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/615896bd67b960440e6761d8b55c4c799760cad9)
b. What is the range?
![{\displaystyle {(-\infty ,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbe6022b6573765e468386496b2a13afc86ac14)
![{\displaystyle {(-\infty ,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbe6022b6573765e468386496b2a13afc86ac14)
c. Where is
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
continuous?
![{\displaystyle {(3/2,10)and(x>10)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/250708f7b2fd04c788499fe75aba0348f5608be8)
![{\displaystyle {(3/2,10)and(x>10)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/250708f7b2fd04c788499fe75aba0348f5608be8)
58. Consider the following function
![{\displaystyle f(x)={\frac {x-7}{x^{2}-49}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37e849bf4d01c50bac417ebf32554448e78e5638)
a. What is the domain?
![{\displaystyle {(-\infty ,-7)\cup (-7,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5bb0c1c0189fb42be8a9b9a479e07ad1e80beb0)
![{\displaystyle {(-\infty ,-7)\cup (-7,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5bb0c1c0189fb42be8a9b9a479e07ad1e80beb0)
b. What is the range?
![{\displaystyle {(-\infty ,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbe6022b6573765e468386496b2a13afc86ac14)
![{\displaystyle {(-\infty ,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbe6022b6573765e468386496b2a13afc86ac14)
c. Where is
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
continuous?
![{\displaystyle {(-\infty ,-7)and(-7,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f1e8b0baf806b053e33757a24beba27f54e899f)
![{\displaystyle {(-\infty ,-7)and(-7,\infty )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f1e8b0baf806b053e33757a24beba27f54e899f)
59. Find the equation of the line that passes through the point (1,-1) and has slope 3.
![{\displaystyle 3x-y=4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c042ddf9241f8b40c310ada85e3dfa3d1ca0be2)
![{\displaystyle 3x-y=4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c042ddf9241f8b40c310ada85e3dfa3d1ca0be2)
60. Find the equation of the line that passes through the origin and the point (2,3).
![{\displaystyle 3x-2y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f490d1d4c020a6204581090c8f9cca0957702651)
![{\displaystyle 3x-2y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f490d1d4c020a6204581090c8f9cca0957702651)