Algebra

Convert to interval notation

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

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

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

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

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

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

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

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

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

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

State the following intervals using set notation

11. ${\displaystyle [3,4]}$

12. ${\displaystyle [3,4)}$

13. ${\displaystyle (3,\infty )}$

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

15. ${\displaystyle \left(-\pi ,{\tfrac {15}{16}}\right)}$

16. ${\displaystyle (-\infty ,\infty )}$

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. ${\displaystyle |x+y|=|x|+|y|}$

18. ${\displaystyle |x+y|\geq |x|+|y|}$

19. ${\displaystyle |x+y|\leq |x|+|y|}$

Evaluate the following expressions

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

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

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

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

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

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

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

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

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

Simplify the following

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

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

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

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

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

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

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

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

Find the roots of the following polynomials

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

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

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

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

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

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

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

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

Factor the following expressions

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

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

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

Simplify the following

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

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

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

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

Functions

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

a. Compute ${\displaystyle f(0)}$ , ${\displaystyle f(2)}$ , and ${\displaystyle f(-1.2)}$ .

b. What are the domain and range of ${\displaystyle f}$ ?

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

53. Let ${\displaystyle f(x)=x+2}$ , ${\displaystyle g(x)=1/x}$ .

a. Give formulae for

i. ${\displaystyle f+g}$

ii. ${\displaystyle f-g}$

iii. ${\displaystyle g-f}$

iv. ${\displaystyle f\times g}$

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

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

vii. ${\displaystyle f\circ g}$

viii. ${\displaystyle g\circ f}$

b. Compute ${\displaystyle f(g(2))}$ and ${\displaystyle g(f(2))}$ .

c. Do ${\displaystyle f}$ and ${\displaystyle g}$ have inverses? If so, find formulae for them.

54. Does this graph represent a function?

55. Consider the following function

${\displaystyle f(x)={\begin{cases}-{\frac {1}{9}}&{\mbox{if }}x<-1\\2&{\mbox{if }}-1\leq x\leq 0\\x+3&{\mbox{if }}x>0\end{cases}}}$

c. Where is ${\displaystyle f}$ continuous?

56. Consider the following function

${\displaystyle f(x)={\begin{cases}x^{2}&{\mbox{if }}x>0\\-1&{\mbox{if }}x\leq 0\end{cases}}}$

c. Where is ${\displaystyle f}$ continuous?

57. Consider the following function

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

c. Where is ${\displaystyle f}$ continuous?

58. Consider the following function

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

c. Where is ${\displaystyle f}$ continuous?

Graphing