Algebra
Convert to interval notation
State the following intervals using set notation
Which one of the following is a true statement?
Hint: the true statement is often referred to as the triangle inequality. Give examples where the other two are false.
17.
![{\displaystyle |x+y|=|x|+|y|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3ec7b9bfa8a71546561942ae229daa107246c7)
18.
![{\displaystyle |x+y|\geq |x|+|y|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5139cd95cc368215289201617a4d54aa16fdff8e)
19.
![{\displaystyle |x+y|\leq |x|+|y|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b02fe24bab89ece41c4640d04bb5e8996ef92ee)
Evaluate the following expressions
Simplify the following
Find the roots of the following polynomials
Factor the following expressions
Simplify the following
Functions
52. Let
.
53. Let
,
.
- a. Give formulae for
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))=2.5\ ,\ g(f(2))=0.25}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6862d53e59541c4036b364b8accd8fa9eae30aae)
![{\displaystyle f(g(2))=2.5\ ,\ g(f(2))=0.25}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6862d53e59541c4036b364b8accd8fa9eae30aae)
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.
![{\displaystyle f^{-1}(x)=x-2\ ,\ g^{-1}(x)={\frac {1}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffcec61f151a05af5adaed5e0c80f05aa3d7e81b)
![{\displaystyle f^{-1}(x)=x-2\ ,\ g^{-1}(x)={\frac {1}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffcec61f151a05af5adaed5e0c80f05aa3d7e81b)
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)
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/4ce2a464fe8e7c6d4500d4d27c62a3b0f71f9419)
a. What is the domain?
![{\displaystyle (-\infty ,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8c11c44279888c9e395eeb5f45d121348ae10a)
![{\displaystyle (-\infty ,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8c11c44279888c9e395eeb5f45d121348ae10a)
b. What is the range?
![{\displaystyle \left(-{\tfrac {1}{9}},\infty \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b8ce0b24f65b966a55f1e7655169b23c57f1c4)
![{\displaystyle \left(-{\tfrac {1}{9}},\infty \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b8ce0b24f65b966a55f1e7655169b23c57f1c4)
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/80d24be5f0eb4a9173da6038badc8659546021d0)
![{\displaystyle x>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0)
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/d44d3fffbdd7e7e53808ed09dadf38860f826530)
a. What is the domain?
![{\displaystyle (-\infty ,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8c11c44279888c9e395eeb5f45d121348ae10a)
![{\displaystyle (-\infty ,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8c11c44279888c9e395eeb5f45d121348ae10a)
b. What is the range?
![{\displaystyle (-1,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dec84bcdac123f01f2ff7ffc65e4feb642b6b576)
![{\displaystyle (-1,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dec84bcdac123f01f2ff7ffc65e4feb642b6b576)
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/80d24be5f0eb4a9173da6038badc8659546021d0)
![{\displaystyle x>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0)
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 (1.5,10)\cup (10,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3be68fb4ed0441ad4f6833f0701831d3c7e0ac98)
![{\displaystyle (1.5,10)\cup (10,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3be68fb4ed0441ad4f6833f0701831d3c7e0ac98)
b. What is the range?
![{\displaystyle (-\infty ,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8c11c44279888c9e395eeb5f45d121348ae10a)
![{\displaystyle (-\infty ,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8c11c44279888c9e395eeb5f45d121348ae10a)
c. Where is
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
continuous?
![{\displaystyle (1.5,10)\cup (10,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3be68fb4ed0441ad4f6833f0701831d3c7e0ac98)
![{\displaystyle (1.5,10)\cup (10,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3be68fb4ed0441ad4f6833f0701831d3c7e0ac98)
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/4049eca5a1bb999e0ca162fe0c6b5a2bc4722566)
![{\displaystyle (-\infty ,-7)\cup (-7,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4049eca5a1bb999e0ca162fe0c6b5a2bc4722566)
b. What is the range?
![{\displaystyle (-\infty ,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8c11c44279888c9e395eeb5f45d121348ae10a)
![{\displaystyle (-\infty ,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c8c11c44279888c9e395eeb5f45d121348ae10a)
c. Where is
![{\displaystyle f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
continuous?
![{\displaystyle (-\infty ,-7)\cup (-7,7)\cup (7,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18c33850e7a86b7b0418cabfc7b97d1201060db8)
![{\displaystyle (-\infty ,-7)\cup (-7,7)\cup (7,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18c33850e7a86b7b0418cabfc7b97d1201060db8)
Graphing
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)
Solutions