# Calculus/Precalculus/Exercises
## Algebra
### Convert to interval notation

$\{x:-4<x<2\}$
$(-4,2)$
$(-4,2)$
$\left\{x:-{\tfrac {7}{3}}\leq x\leq -{\tfrac {1}{3}}\right\}$
$\left[-{\tfrac {7}{3}},-{\tfrac {1}{3}}\right]$
$\left[-{\tfrac {7}{3}},-{\tfrac {1}{3}}\right]$
$\{x:-\pi \leq x<\pi \}$
$[-\pi ,\pi )$
$[-\pi ,\pi )$
$\{x:x\leq {\tfrac {17}{9}}\}$
$\left(-\infty ,{\tfrac {17}{9}}\right]$
$\left(-\infty ,{\tfrac {17}{9}}\right]$
$\{x:5\leq x+1\leq 6\}$
$[4,5]$
$[4,5]$
$\left\{x:x-{\tfrac {1}{4}}<1\right\}$
$\left(-\infty ,{\frac {5}{4}}\right)$
$\left(-\infty ,{\frac {5}{4}}\right)$
$\{x:3>3x\}$
$(-\infty ,1)$
$(-\infty ,1)$
$\{x:0\leq 2x+1<3\}$
$\left[-{\tfrac {1}{2}},1\right)$
$\left[-{\tfrac {1}{2}},1\right)$
$\{x:5<x{\text{ and }}x<6\}$
$(5,6)$
$(5,6)$
$\{x:5<x{\text{ or }}x<6\}$
$(-\infty ,\infty )$
$(-\infty ,\infty )$

### State the following intervals using set notation

$[3,4]$
$\{x:3\leq x\leq 4\}$
$\{x:3\leq x\leq 4\}$
$[3,4)$
$\{x:3\leq x<4\}$
$\{x:3\leq x<4\}$
$(3,\infty )$
$\{x:x>3\}$
$\{x:x>3\}$
$\left(-{\tfrac {1}{3}},{\tfrac {1}{3}}\right)$
$\left\{x:-{\tfrac {1}{3}}<x<{\tfrac {1}{3}}\right\}$
$\left\{x:-{\tfrac {1}{3}}<x<{\tfrac {1}{3}}\right\}$
$\left(-\pi ,{\tfrac {15}{16}}\right)$
$\left\{x:-\pi <x<{\tfrac {15}{16}}\right\}$
$\left\{x:-\pi <x<{\tfrac {15}{16}}\right\}$
$(-\infty ,\infty )$
$\{x:x\in \mathbb {R} \}$
$\{x:x\in \mathbb {R} \}$

### 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.
$|x+y|=|x|+|y|$
$|x+y|\geq |x|+|y|$
$|x+y|\leq |x|+|y|$

### Evaluate the following expressions

$8^{\frac {1}{3}}$
$2$
$2$
$(-8)^{\frac {1}{3}}$
$-2$
$-2$
$\left({\frac {1}{8}}\right)^{\frac {1}{3}}$
${\frac {1}{2}}$
${\frac {1}{2}}$
$\left(8^{\frac {2}{3}}\right)\left(8^{\frac {3}{2}}\right)(8^{0})$
$8^{\frac {13}{6}}$
$8^{\frac {13}{6}}$
$\left(\left({\frac {1}{8}}\right)^{\frac {1}{3}}\right)^{7}$
${\frac {1}{128}}$
${\frac {1}{128}}$
${\sqrt[{3}]{\frac {27}{8}}}$
${\frac {3}{2}}$
${\frac {3}{2}}$
${\frac {4^{5}\cdot 4^{-2}}{4^{3}}}$
$1$
$1$
$\left({\sqrt {27}}\right)^{\frac {2}{3}}$
$3$
$3$
${\frac {\sqrt {27}}{\sqrt[{3}]{9}}}$
$3^{\frac {5}{6}}$
$3^{\frac {5}{6}}$

### Simplify the following

$x^{3}+3x^{3}$
$4x^{3}$
$4x^{3}$
${\frac {x^{3}+3x^{3}}{x^{2}}}$
$4x$
$4x$
$(x^{3}+3x^{3})^{3}$
$64x^{9}$
$64x^{9}$
${\frac {x^{15}+x^{3}}{x}}$
$x^{14}+x^{2}$
$x^{14}+x^{2}$
$(2x^{2})(3x^{-2})$
$6$
$6$
${\frac {x^{2}y^{-3}}{x^{3}y^{2}}}$
${\frac {1}{xy^{5}}}$
${\frac {1}{xy^{5}}}$
${\sqrt {x^{2}y^{4}}}$
$|xy^{2}|$
$|xy^{2}|$
$\left({\frac {8x^{6}}{y^{4}}}\right)^{\frac {1}{3}}$
${\frac {2x^{2}}{y^{\frac {4}{3}}}}$
${\frac {2x^{2}}{y^{\frac {4}{3}}}}$

### Find the roots of the following polynomials

$x^{2}-1$
$x=\pm 1$
$x=\pm 1$
$x^{2}+2x+1$
$x=-1$
$x=-1$
$x^{2}+7x+12$
$x=-3,x=-4$
$x=-3,x=-4$
$3x^{2}-5x-2$
$x=2,x=-{\frac {1}{3}}$
$x=2,x=-{\frac {1}{3}}$
$x^{2}+{\frac {5}{6}}x+{\frac {1}{6}}$
$x=-{\frac {1}{3}},x=-{\frac {1}{2}}$
$x=-{\frac {1}{3}},x=-{\frac {1}{2}}$
$4x^{3}+4x^{2}+x$
$x=0,x=-{\frac {1}{2}}$
$x=0,x=-{\frac {1}{2}}$
$x^{4}-1$
$x=\pm i,x=\pm 1$
$x=\pm i,x=\pm 1$
$x^{3}+2x^{2}-4x-8$
$x=\pm 2$
$x=\pm 2$

### Factor the following expressions

$4a^{2}-ab-3b^{2}$
$(4a+3b)(a-b)$
$(4a+3b)(a-b)$
$(c+d)^{2}-4$
$(c+d+2)(c+d-2)$
$(c+d+2)(c+d-2)$
$4x^{2}-9y^{2}$
$(2x+3y)(2x-3y)$
$(2x+3y)(2x-3y)$

### Simplify the following

${\frac {x^{2}-1}{x+1}}$
$x-1,x\neq -1$
$x-1,x\neq -1$
${\frac {3x^{2}+4x+1}{x+1}}$
$3x+1,x\neq -1$
$3x+1,x\neq -1$
${\frac {4x^{2}-9}{4x^{2}+12x+9}}$
${\frac {2x-3}{2x+3}}$
${\frac {2x-3}{2x+3}}$
${\frac {x^{2}+y^{2}+2xy}{x(x+y)}}$
${\frac {x+y}{x}},x\neq -y$
${\frac {x+y}{x}},x\neq -y$

## Functions

52. Let $f(x)=x^{2}$ .
$f(0)$
$f(2)$
$f(-1.2)$
$f(0)=0$
$f(2)=4$
$f(-1.2)=1.44$
$f(0)=0$
$f(2)=4$
$f(-1.2)=1.44$
$f$
$(-\infty ,\infty )$
$[0,\infty )$
$(-\infty ,\infty )$
$[0,\infty )$
$f$
$f$
$x=1$
$x=-1$
$f(x)=1$
$f$
$x=1$
$x=-1$
$f(x)=1$
53. Let $f(x)=x+2$ , $g(x)=1/x$ .
    a. Give formulae for
$f+g$
$(f+g)(x)=x+2+{\frac {1}{x}}$
$(f+g)(x)=x+2+{\frac {1}{x}}$
$f-g$
$(f-g)(x)=x+2-{\frac {1}{x}}$
$(f-g)(x)=x+2-{\frac {1}{x}}$
$g-f$
$(g-f)(x)={\frac {1}{x}}-x-2$
$(g-f)(x)={\frac {1}{x}}-x-2$
$f\times g$
$(f\times g)(x)=1+{\frac {2}{x}}$
$(f\times g)(x)=1+{\frac {2}{x}}$
${\frac {f}{g}}$
$\left({\frac {f}{g}}\right)(x)=x^{2}+2x$
$\left({\frac {f}{g}}\right)(x)=x^{2}+2x$
${\frac {g}{f}}$
$\left({\frac {g}{f}}\right)(x)={\frac {1}{x^{2}+2x}}$
$\left({\frac {g}{f}}\right)(x)={\frac {1}{x^{2}+2x}}$
$f\circ g$
$(f\circ g)(x)={\frac {1}{x}}+2$
$(f\circ g)(x)={\frac {1}{x}}+2$
$g\circ f$
$(g\circ f)(x)={\frac {1}{x+2}}$
$(g\circ f)(x)={\frac {1}{x+2}}$
$f(g(2))$
$g(f(2))$
$f(g(2))=2.5\ ,\ g(f(2))=0.25$
$f(g(2))=2.5\ ,\ g(f(2))=0.25$
$f$
$g$
$f^{-1}(x)=x-2\ ,\ g^{-1}(x)={\frac {1}{x}}$
$f^{-1}(x)=x-2\ ,\ g^{-1}(x)={\frac {1}{x}}$
55. Consider the following function
    $f(x)={\begin{cases}-{\frac {1}{9}}&{\mbox{if }}x<-1\\2&{\mbox{if }}-1\leq x\leq 0\\x+3&{\mbox{if }}x>0\end{cases}}$
$(-\infty ,\infty )$
$(-\infty ,\infty )$
$\left(-{\tfrac {1}{9}},\infty \right)$
$\left(-{\tfrac {1}{9}},\infty \right)$
$f$
$x>0$
$x>0$
56. Consider the following function
    $f(x)={\begin{cases}x^{2}&{\mbox{if }}x>0\\-1&{\mbox{if }}x\leq 0\end{cases}}$
$(-\infty ,\infty )$
$(-\infty ,\infty )$
$(-1,\infty )$
$(-1,\infty )$
$f$
$x>0$
$x>0$
57. Consider the following function
    $f(x)={\frac {\sqrt {2x-3}}{x-10}}$
$(1.5,10)\cup (10,\infty )$
$(1.5,10)\cup (10,\infty )$
$(-\infty ,\infty )$
$(-\infty ,\infty )$
$f$
$(1.5,10)\cup (10,\infty )$
$(1.5,10)\cup (10,\infty )$
58. Consider the following function
    $f(x)={\frac {x-7}{x^{2}-49}}$
$(-\infty ,-7)\cup (-7,\infty )$
$(-\infty ,-7)\cup (-7,\infty )$
$(-\infty ,\infty )$
$(-\infty ,\infty )$
$f$
$(-\infty ,-7)\cup (-7,7)\cup (7,\infty )$
$(-\infty ,-7)\cup (-7,7)\cup (7,\infty )$

## Graphing

$3x-y=4$
$3x-y=4$
$3x-2y=0$
$3x-2y=0$
Solutions
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