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?

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?

b. What is the range?

c. Where is

f

continuous?

57. Consider the following function

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