Calculus/Precalculus/Exercises

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	Precalculus/Exercises

[edit] Algebra

[edit] Convert to interval notation

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

$[ - π,π)$

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

$(-\infty,\infty)$

[edit] 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. $(-\frac{1}{3}, \frac{1}{3}) \,$

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

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

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

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

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

[edit] 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

[edit] 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. $(8^{2/3}) (8^{3/2}) (8^0) \,$

$2 13 / 6$

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

$3 5 / 6$

[edit] Simplify the following

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

$4 x 3$

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

$4 x$

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

$64 x 9$

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

$x y 2$

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

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

[edit] 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. $x^3 + 2x^2 - 4x - 8 \,$

$x=\pm2$

[edit] Factor the following expressions

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

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

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

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

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

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

[edit] 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. $\frac{x^2 + y^2 +2xy}{x(x+y)} \,$

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

[edit] Functions

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

a. Compute

f (0)

and

f (2)

.

b. What are the domain and range of

f

?

c. Does

f

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

53. Let $f (x) = x + 2$ , $g (x) = 1 / x$ .

a. Give formulae for

i.

f + g

ii.

f - g

iii.

g - f

iv. $f\times g$

v.

f / g

vi.

g / f

vii. $f\circ g$

viii. $g\circ f$

b. Compute

f (g (2))

and

g (f (2))

.

c. Do

f

and

g

have inverses? If so, find formulae for them.

54. Does this graph represent a function?

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

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