Calculus/Precalculus/Exercises

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Precalculus/Exercises

Algebra[edit]

Convert to interval notation[edit]

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 notation[edit]

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 expressions[edit]

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 following[edit]

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 polynomials[edit]

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 expressions[edit]

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 following[edit]

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

Functions[edit]

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

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".

Graphing[edit]

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