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), f(2), and f(-1.2).

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)

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.

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?

{(-1/9,\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?

{(-\infty,\infty)}

b. What is the range?

{(-1, \infty)}

c. Where is f continuous?

{x>0}

57. Consider the following function

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

{(3/2,10) \cup (10,\infty)}

b. What is the range?

{(-\infty,\infty)}

c. Where is f continuous?

{(3/2,10) and (x>10)}

58. Consider the following function

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

{(-\infty,-7) \cup (-7,\infty)}

b. What is the range?

{(-\infty,\infty)}

c. Where is f continuous?

{(-\infty,-7) and (-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

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