Last modified on 24 October 2014, at 16:16

Calculus/Precalculus/Exercises

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Precalculus/Exercises

AlgebraEdit

Convert to interval notationEdit

1.  \{x:-4<x<2\} \,

(-4,2)

2.  \{x:-\frac{7}{3} \leq x \leq -\frac{1}{3}\}

[-\frac{7}{3},-\frac{1}{3}]

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

[-\pi,\pi)

4.  \{x:x \leq \frac{17}{9}\}

(-\infty, \frac{17}{9}]

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

[4, 5]

6.  \{x:x - \frac{1}{4} < 1\} \,

(-\infty, \frac{5}{4})

7.  \{x:3 > 3x\} \,

(-\infty, 1)

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

[-\frac{1}{2}, 1)

9.  \{x:5<x \mbox{ and } x<6\} \,

(5,6)

10.  \{x:5<x \mbox{ or } x<6\} \,

(-\infty,\infty)

State the following intervals using set notationEdit

11.  [3,4] \,

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

12.  [3,4) \,

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

13.  (3,\infty)

\{x:x>3\}

14.  (-\frac{1}{3}, \frac{1}{3}) \,

\{x:-\frac{1}{3}<x<\frac{1}{3}\}

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

\{x:-\pi<x<\frac{15}{16}\}

16.  (-\infty,\infty)

\{x:x\in\Re\}

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 expressionsEdit

20.  8^{1/3} \,

2

21.  (-8)^{1/3} \,

-2

22.  \bigg(\frac{1}{8}\bigg)^{1/3} \,

\frac{1}{2}

23.  (8^{2/3}) (8^{3/2}) (8^0) \,

8^{13/6}

24.  \bigg( \bigg(\frac{1}{8}\bigg)^{1/3} \bigg)^7

\frac{1}{128}

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

\frac{3}{2}

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

1

27.  \bigg(\sqrt{27}\bigg)^{2/3}

3

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

3^{5/6}

Simplify the followingEdit

29.  x^3 + 3x^3 \,

4x^3

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

4x

31.  (x^3+3x^3)^3 \,

64x^9

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

x^{14}+x^2

33.  (2x^2)(3x^{-2}) \,

6

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

\frac{1}{xy^5}

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

xy^2

36.  \bigg(\frac{8x^6}{y^4}\bigg)^{1/3}

\frac{2x^2}{y^{4/3}}

Find the roots of the following polynomialsEdit

37.  x^2 - 1 \,

x=\pm1

38.  x^2 +2x +1 \,

x=-1

39.  x^2 + 7x + 12 \,

x=-3, x=-4

40.  3x^2 - 5x -2 \,

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

41.  x^2 + 5/6x + 1/6 \,

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

42.  4x^3 + 4x^2 + x \,

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

43.  x^4 - 1 \,

x=\pm i, x=\pm 1

44.  x^3 + 2x^2 - 4x - 8 \,

x=\pm2

Factor the following expressionsEdit

45.  4a^2 - ab - 3b^2 \,

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

46.  (c+d)^2 - 4 \,

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

47.  4x^2 - 9y^2 \,

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

Simplify the followingEdit

48.  \frac{x^2 -1}{x+1} \,

x-1, x\neq-1

49.  \frac{3x^2 + 4x + 1}{x+1} \,

3x+1, x\neq-1

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

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

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

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

FunctionsEdit

52. Let f(x)=x^2.

a. Compute f(0) and f(2).

{0,4}

b. What are the domain and range of f?

{(-\infty,\infty)}

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

{x^{1/2}}

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}

ii. f-g

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

iii. g-f

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

iv. f\times g

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

v. f/g

(f / g)(x) = x^2 + 2x

vi. g/f

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

vii. f\circ g

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

viii. g\circ f

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

b. Compute f(g(2)) and g(f(2)).

f(g(2))=5/2, g(f(2))=1/4

c. Do f and g have inverses? If so, find formulae for them.

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

54. Does this graph represent a function? Sinx over x.svg

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)}

b. What is the range?

{(-\infty,\infty)}

c. Where is f continuous?

{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?
b. What is the range?
c. Where is f continuous?

57. Consider the following function

f(x) = \frac{\sqrt{2x-3}}{x-10}
a. What is the domain?

When you find the answer, you can add it here by clicking "edit".

b. What is the range?

When you find the answer, you can add it here by clicking "edit".

c. Where is f continuous?

When you find the answer, you can add it here by clicking "edit".

58. Consider the following function

f(x) = \frac{x-7}{x^2-49}
a. What is the domain?

When you find the answer, you can add it here by clicking "edit".

b. What is the range?

When you find the answer, you can add it here by clicking "edit".

c. Where is f continuous?

When you find the answer, you can add it here by clicking "edit".

GraphingEdit

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

3x-y=4

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

3x-2y=0

Solutions

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Precalculus/Exercises