Calculus/Precalculus/Exercises

Contents

AlgebraEdit

Convert to interval notationEdit

1. $\{x:-4<x<2\}$$\{x:-4<x<2\}$

2. $\left\{x:-{\tfrac {7}{3}}\leq x\leq -{\tfrac {1}{3}}\right\}$$\left\{x:-{\tfrac {7}{3}}\leq x\leq -{\tfrac {1}{3}}\right\}$

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

4. $\{x:x\leq {\tfrac {17}{9}}\}$$\{x:x\leq {\tfrac {17}{9}}\}$

5. $\{x:5\leq x+1\leq 6\}$$\{x:5\leq x+1\leq 6\}$

6. $\left\{x:x-{\tfrac {1}{4}}<1\right\}$$\left\{x:x-{\tfrac {1}{4}}<1\right\}$

7. $\{x:3>3x\}$$\{x:3>3x\}$

8. $\{x:0\leq 2x+1<3\}$$\{x:0\leq 2x+1<3\}$

9. $\{x:5<x{\mbox{ and }}x<6\}$$\{x:5<x{\mbox{ and }}x<6\}$

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

State the following intervals using set notationEdit

11. $[3,4]$$[3,4]$

12. $[3,4)$$[3,4)$

13. $(3,\infty )$$(3,\infty )$

14. $\left(-{\tfrac {1}{3}},{\tfrac {1}{3}}\right)$$\left(-{\tfrac {1}{3}},{\tfrac {1}{3}}\right)$

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

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

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|$$|x+y|=|x|+|y|$

18. $|x+y|\geq |x|+|y|$$|x+y|\geq |x|+|y|$

19. $|x+y|\leq |x|+|y|$$|x+y|\leq |x|+|y|$

Evaluate the following expressionsEdit

20. $8^{\frac {1}{3}}$$8^{\frac {1}{3}}$

21. $(-8)^{\frac {1}{3}}$$(-8)^{\frac {1}{3}}$

22. $\left({\frac {1}{8}}\right)^{\frac {1}{3}}$$\left({\frac {1}{8}}\right)^{\frac {1}{3}}$

23. $(8^{\frac {2}{3}})(8^{\frac {3}{2}})(8^{0})$$(8^{\frac {2}{3}})(8^{\frac {3}{2}})(8^{0})$

24. $\left(\left({\frac {1}{8}}\right)^{\frac {1}{3}}\right)^{7}$$\left(\left({\frac {1}{8}}\right)^{\frac {1}{3}}\right)^{7}$

25. ${\sqrt[{3}]{\frac {27}{8}}}$${\sqrt[{3}]{\frac {27}{8}}}$

26. ${\frac {4^{5}\cdot 4^{-2}}{4^{3}}}$${\frac {4^{5}\cdot 4^{-2}}{4^{3}}}$

27. $\left({\sqrt {27}}\right)^{\frac {2}{3}}$$\left({\sqrt {27}}\right)^{\frac {2}{3}}$

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

Simplify the followingEdit

29. $x^{3}+3x^{3}$$x^{3}+3x^{3}$

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

31. $(x^{3}+3x^{3})^{3}$$(x^{3}+3x^{3})^{3}$

32. ${\frac {x^{15}+x^{3}}{x}}$${\frac {x^{15}+x^{3}}{x}}$

33. $(2x^{2})(3x^{-2})$$(2x^{2})(3x^{-2})$

34. ${\frac {x^{2}y^{-3}}{x^{3}y^{2}}}$${\frac {x^{2}y^{-3}}{x^{3}y^{2}}}$

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

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

Find the roots of the following polynomialsEdit

37. $x^{2}-1$$x^{2}-1$

38. $x^{2}+2x+1$$x^{2}+2x+1$

39. $x^{2}+7x+12$$x^{2}+7x+12$

40. $3x^{2}-5x-2$$3x^{2}-5x-2$

41. $x^{2}+{\frac {5}{6}}x+{\frac {1}{6}}$$x^{2}+{\frac {5}{6}}x+{\frac {1}{6}}$

42. $4x^{3}+4x^{2}+x$$4x^{3}+4x^{2}+x$

43. $x^{4}-1$$x^{4}-1$

44. $x^{3}+2x^{2}-4x-8$$x^{3}+2x^{2}-4x-8$

Factor the following expressionsEdit

45. $4a^{2}-ab-3b^{2}$$4a^{2}-ab-3b^{2}$

46. $(c+d)^{2}-4$$(c+d)^{2}-4$

47. $4x^{2}-9y^{2}$$4x^{2}-9y^{2}$

Simplify the followingEdit

48. ${\frac {x^{2}-1}{x+1}}$${\frac {x^{2}-1}{x+1}}$

49. ${\frac {3x^{2}+4x+1}{x+1}}$${\frac {3x^{2}+4x+1}{x+1}}$

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

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

FunctionsEdit

52. Let $f(x)=x^{2}$$f(x)=x^{2}$ .

a. Compute $f(0)$$f(0)$ , $f(2)$$f(2)$ , and $f(-1.2)$$f(-1.2)$ .

b. What are the domain and range of $f$$f$ ?

c. Does $f$$f$ have an inverse? If so, find a formula for it.

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

a. Give formulae for

i. $f+g$$f+g$

ii. $f-g$$f-g$

iii. $g-f$$g-f$

iv. $f\times g$$f\times g$

v. ${\frac {f}{g}}$${\frac {f}{g}}$

vi. ${\frac {g}{f}}$${\frac {g}{f}}$

vii. $f\circ g$$f\circ g$

viii. $g\circ f$$g\circ f$

b. Compute $f(g(2))$$f(g(2))$ and $g(f(2))$$g(f(2))$ .

c. Do $f$$f$ and $g$$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}}$$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$$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}}$$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$$f$ continuous?

57. Consider the following function

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

a. What is the domain?

b. What is the range?

c. Where is $f$$f$ continuous?

58. Consider the following function

$f(x)={\frac {x-7}{x^{2}-49}}$$f(x)={\frac {x-7}{x^{2}-49}}$

a. What is the domain?

b. What is the range?

c. Where is $f$$f$ continuous?

GraphingEdit

59. Find the equation of the line that passes through the point (1,-1) and has slope 3.

60. Find the equation of the line that passes through the origin and the point (2,3).