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	Precalculus/Exercises

Algebra

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

$[-\pi,\pi)$

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

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?

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. $\frac{3x^3 + 4x + 1}{x+1} \,$

$3x+1, x\neq-1$

50. $\frac{4x^2 - 9}{4x^2 + 12x + 9} \,$

$\frac{2x-3}{2x+3}$

51. $\frac{x^2 + y^2 +2xy}{x(x+y)} \,$

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

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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.

$f^{-1}(x)=x-1, g^{-1}(x)=\frac{1}{x}$

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?

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

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