Calculus/Precalculus/Exercises

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	Precalculus/Exercises

Algebra[edit | edit source]

Convert to interval notation[edit | edit source]

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{\text{ and }}x<6\}

(5,6)

(5,6)

10.

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

(-\infty ,\infty )

(-\infty ,\infty )

State the following intervals using set notation[edit | edit source]

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.

\left(-\pi ,{\tfrac {15}{16}}\right)

\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 | edit source]

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 | edit source]

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.

\left(8^{\frac {2}{3}}\right)\left(8^{\frac {3}{2}}\right)(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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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 | edit source]

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

a. Compute

f(0)

,

f(2)

, and

f(-1.2)

.

f(0)=0

,

f(2)=4

, and

f(-1.2)=1.44

f(0)=0

,

f(2)=4

, and

f(-1.2)=1.44

b. What are the domain and range of

f

?

Domain is

(-\infty ,\infty )

; range is

[0,\infty )

Domain is

(-\infty ,\infty )

; range is

[0,\infty )

c. Does

f

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

No,

f

is not one-to-one. For example, both

x=1

and

x=-1

result in

f(x)=1

.

No,

f

is not one-to-one. For example, both

x=1

and

x=-1

result in

f(x)=1

.

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

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 | edit source]

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 | edit source]

Convert to interval notation[edit | edit source]

State the following intervals using set notation[edit | edit source]

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

Evaluate the following expressions[edit | edit source]

Simplify the following[edit | edit source]

Find the roots of the following polynomials[edit | edit source]

Factor the following expressions[edit | edit source]

Simplify the following[edit | edit source]

Functions[edit | edit source]

Graphing[edit | edit source]

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