Calculus/Precalculus/Exercises

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	Precalculus/Exercises

Algebra[edit]

Convert to interval notation[edit]

1.

\{x:-4<x<2\}

$(-4,2)$ $(-4,2)$

2.

\left\{x:-{\tfrac {7}{3}}\leq x\leq -{\tfrac {1}{3}}\right\}

$\left[-{\tfrac {7}{3}},-{\tfrac {1}{3}}\right]$ $\left[-{\tfrac {7}{3}},-{\tfrac {1}{3}}\right]$

3.

\{x:-\pi \leq x<\pi \}

$[-\pi ,\pi )$ $[-\pi ,\pi )$

4.

\{x:x\leq {\tfrac {17}{9}}\}

$\left(-\infty ,{\tfrac {17}{9}}\right]$ $\left(-\infty ,{\tfrac {17}{9}}\right]$

5.

\{x:5\leq x+1\leq 6\}

$[4,5]$ $[4,5]$

6.

\left\{x:x-{\tfrac {1}{4}}<1\right\}

$\left(-\infty ,{\frac {5}{4}}\right)$ $\left(-\infty ,{\frac {5}{4}}\right)$

7.

\{x:3>3x\}

$(-\infty ,1)$ $(-\infty ,1)$

8.

\{x:0\leq 2x+1<3\}

$\left[-{\tfrac {1}{2}},1\right)$ $\left[-{\tfrac {1}{2}},1\right)$

9.

\{x:5<x{\mbox{ and }}x<6\}

$(5,6)$ $(5,6)$

10.

\{x:5<x{\mbox{ or }}x<6\}

$(-\infty ,\infty )$ $(-\infty ,\infty )$

State the following intervals using set notation[edit]

11.

[3,4]

$\{x:3\leq x\leq 4\}$ $\{x:3\leq x\leq 4\}$

12.

[3,4)

$\{x:3\leq x<4\}$ $\{x:3\leq x<4\}$

13.

(3,\infty )

$\{x:x>3\}$ $\{x:x>3\}$

14.

\left(-{\tfrac {1}{3}},{\tfrac {1}{3}}\right)

$\left\{x:-{\tfrac {1}{3}}<x<{\tfrac {1}{3}}\right\}$ $\left\{x:-{\tfrac {1}{3}}<x<{\tfrac {1}{3}}\right\}$

15.

(-\pi ,{\frac {15}{16}})

$\left\{x:-\pi <x<{\tfrac {15}{16}}\right\}$ $\left\{x:-\pi <x<{\tfrac {15}{16}}\right\}$

16.

(-\infty ,\infty )

$\{x:x\in \mathbb {R} \}$ $\{x:x\in \mathbb {R} \}$

Which one of the following is a true statement?[edit]

Hint: the true statement is often referred to as the triangle inequality. Give examples where the other two are false.

17.

|x+y|=|x|+|y|

false

18.

|x+y|\geq |x|+|y|

false

19.

|x+y|\leq |x|+|y|

true

Evaluate the following expressions[edit]

20.

8^{\frac {1}{3}}

$2$ $2$

21.

(-8)^{\frac {1}{3}}

$-2$ $-2$

22.

\left({\frac {1}{8}}\right)^{\frac {1}{3}}

${\frac {1}{2}}$ ${\frac {1}{2}}$

23.

(8^{\frac {2}{3}})(8^{\frac {3}{2}})(8^{0})

$8^{\frac {13}{6}}$ $8^{\frac {13}{6}}$

24.

\left(\left({\frac {1}{8}}\right)^{\frac {1}{3}}\right)^{7}

${\frac {1}{128}}$ ${\frac {1}{128}}$

25.

{\sqrt[{3}]{\frac {27}{8}}}

${\frac {3}{2}}$ ${\frac {3}{2}}$

26.

{\frac {4^{5}\cdot 4^{-2}}{4^{3}}}

$1$ $1$

27.

\left({\sqrt {27}}\right)^{\frac {2}{3}}

$3$ $3$

28.

{\frac {\sqrt {27}}{\sqrt[{3}]{9}}}

$3^{\frac {5}{6}}$ $3^{\frac {5}{6}}$

Simplify the following[edit]

29.

x^{3}+3x^{3}

$4x^{3}$ $4x^{3}$

30.

{\frac {x^{3}+3x^{3}}{x^{2}}}

$4x$ $4x$

31.

(x^{3}+3x^{3})^{3}

$64x^{9}$ $64x^{9}$

32.

{\frac {x^{15}+x^{3}}{x}}

$x^{14}+x^{2}$ $x^{{14}}+x^{2}$

33.

(2x^{2})(3x^{-2})

$6$ $6$

34.

{\frac {x^{2}y^{-3}}{x^{3}y^{2}}}

${\frac {1}{xy^{5}}}$ ${\frac {1}{xy^{5}}}$

35.

{\sqrt {x^{2}y^{4}}}

$|xy^{2}|$ $|xy^{2}|$

36.

\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[edit]

37.

x^{2}-1

$x=\pm 1$ $x=\pm 1$

38.

x^{2}+2x+1

$x=-1$ $x=-1$

39.

x^{2}+7x+12

$x=-3,x=-4$ $x=-3,x=-4$

40.

3x^{2}-5x-2

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

41.

x^{2}+{\frac {5}{6}}x+{\frac {1}{6}}

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

42.

4x^{3}+4x^{2}+x

$x=0,x=-{\frac {1}{2}}$ $x=0,x=-{\frac {1}{2}}$

43.

x^{4}-1

$x=\pm i,x=\pm 1$ $x=\pm i,x=\pm 1$

44.

x^{3}+2x^{2}-4x-8

$x=\pm 2$ $x=\pm 2$

Factor the following expressions[edit]

45.

4a^{2}-ab-3b^{2}

$(4a+3b)(a-b)$ $(4a+3b)(a-b)$

46.

(c+d)^{2}-4

$(c+d+2)(c+d-2)$ $(c+d+2)(c+d-2)$

47.

4x^{2}-9y^{2}

$(2x+3y)(2x-3y)$ $(2x+3y)(2x-3y)$

Simplify the following[edit]

48.

{\frac {x^{2}-1}{x+1}}

$x-1,x\neq -1$ $x-1,x\neq -1$

49.

{\frac {3x^{2}+4x+1}{x+1}}

$3x+1,x\neq -1$ $3x+1,x\neq -1$

50.

{\frac {4x^{2}-9}{4x^{2}+12x+9}}

${\frac {2x-3}{2x+3}}$ ${\frac {2x-3}{2x+3}}$

51.

{\frac {x^{2}+y^{2}+2xy}{x(x+y)}}

${\frac {x+y}{x}},x\neq -y$ ${\frac {x+y}{x}},x\neq -y$

Functions[edit]

52. Let $f(x)=x^{2}$ $f(x)=x^{2}$ .

a. Compute

f(0)

,

f(2)

, and

f(-1.2)

.

$f(0)=0$ $f(0)=0$ , $f(2)=4$ $f(2)=4$ , and $f(-1.2)=1.44$ $f(-1.2)=1.44$

b. What are the domain and range of

f

?

Domain is $(-\infty ,\infty )$ $(-\infty ,\infty )$ ; range is $[0,\infty )$ $[0,\infty )$

c. Does

f

have an inverse? If so, find a formula for it.

No, $f$ $f$ is not one-to-one. For example, both $x=1$ $x=1$ and $x=-1$ $x=-1$ result in $f(x)=1$ $f(x)=1$ .

53. Let $f(x)=x+2$ $f(x)=x+2$ , $g(x)=1/x$ $g(x)=1/x$ .

a. Give formulae for

i.

f+g

$(f+g)(x)=x+2+{\frac {1}{x}}$ $(f+g)(x)=x+2+{\frac {1}{x}}$

ii.

f-g

$(f-g)(x)=x+2-{\frac {1}{x}}$ $(f-g)(x)=x+2-{\frac {1}{x}}$

iii.

g-f

$(g-f)(x)={\frac {1}{x}}-x-2$ $(g-f)(x)={\frac {1}{x}}-x-2$

iv.

f\times g

$(f\times g)(x)=1+{\frac {2}{x}}$ $(f\times g)(x)=1+{\frac {2}{x}}$

v.

{\frac {f}{g}}

$\left({\frac {f}{g}}\right)(x)=x^{2}+2x$ $\left({\frac {f}{g}}\right)(x)=x^{2}+2x$

vi.

{\frac {g}{f}}

$\left({\frac {g}{f}}\right)(x)={\frac {1}{x^{2}+2x}}$ $\left({\frac {g}{f}}\right)(x)={\frac {1}{x^{2}+2x}}$

vii.

f\circ g

$(f\circ g)(x)={\frac {1}{x}}+2$ $(f\circ g)(x)={\frac {1}{x}}+2$

viii.

g\circ f

$(g\circ f)(x)={\frac {1}{x+2}}$ $(g\circ f)(x)={\frac {1}{x+2}}$

b. Compute

f(g(2))

and

g(f(2))

.

$f(g(2))=2.5\ ,\ g(f(2))=0.25$ $f(g(2))=2.5\ ,\ g(f(2))=0.25$

c. Do

f

and

g

have inverses? If so, find formulae for them.

$f^{-1}(x)=x-2\ ,\ g^{-1}(x)={\frac {1}{x}}$ $f^{-1}(x)=x-2\ ,\ g^{-1}(x)={\frac {1}{x}}$

54. Does this graph represent a function?

Yes.

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}}

a. What is the domain?

$(-\infty ,\infty )$ $(-\infty ,\infty )$

b. What is the range?

$\left(-{\tfrac {1}{9}},\infty \right)$ $\left(-{\tfrac {1}{9}},\infty \right)$

c. Where is

f

continuous?

$x>0$ $x>0$

56. Consider the following function

f(x)={\begin{cases}x^{2}&{\mbox{if }}x>0\\-1&{\mbox{if }}x\leq 0\end{cases}}

a. What is the domain?

$(-\infty ,\infty )$ $(-\infty ,\infty )$

b. What is the range?

$(-1,\infty )$ $(-1,\infty )$

c. Where is

f

continuous?

$x>0$ $x>0$

57. Consider the following function

f(x)={\frac {\sqrt {2x-3}}{x-10}}

a. What is the domain?

$(1.5,10)\cup (10,\infty )$ $(1.5,10)\cup (10,\infty )$

b. What is the range?

$(-\infty ,\infty )$ $(-\infty ,\infty )$

c. Where is

f

continuous?

$(1.5,10)\cup (10,\infty )$ $(1.5,10)\cup (10,\infty )$

58. Consider the following function

f(x)={\frac {x-7}{x^{2}-49}}

a. What is the domain?

$(-\infty ,-7)\cup (-7,\infty )$ $(-\infty ,-7)\cup (-7,\infty )$

b. What is the range?

$(-\infty ,\infty )$ $(-\infty ,\infty )$

c. Where is

f

continuous?

$(-\infty ,-7)\cup (-7,7)\cup (7,\infty )$ $(-\infty ,-7)\cup (-7,7)\cup (7,\infty )$

Graphing[edit]

59. Find the equation of the line that passes through the point (1,-1) and has slope 3.

$3x-y=4$ $3x-y=4$

60. Find the equation of the line that passes through the origin and the point (2,3).

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

Solutions

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	Precalculus/Exercises

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Calculus/Precalculus/Exercises

Contents

Algebra[edit]

Convert to interval notation[edit]

State the following intervals using set notation[edit]

Which one of the following is a true statement?[edit]

Evaluate the following expressions[edit]

Simplify the following[edit]

Find the roots of the following polynomials[edit]

Factor the following expressions[edit]

Simplify the following[edit]

Functions[edit]

Graphing[edit]

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