# Calculus/Precalculus/Exercises

 ← Graphing linear functions Calculus Limits → Precalculus/Exercises

## Algebra

### Convert to interval notation

1. $\{x:-4

$(-4,2)$

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

$(5,6)$

10. $\{x:5

$(-\infty,\infty)$

### State the following intervals using set notation

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}

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

$\{x:-\pi

16. $(-\infty,\infty)$

$\{x:x\in\Re\}$

### Which one of the following is a true statement?

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

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

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

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

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

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

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?

${(-\infty,\infty)}$

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".

## Graphing

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

 ← Graphing linear functions Calculus Limits → Precalculus/Exercises