# Calculus/Precalculus/Exercises

 ← Graphing linear functions Calculus Hyperbolic logarithm and angles → Precalculus/Exercises

## Algebra

### Convert to interval notation

1. ${\displaystyle \{x:-4
${\displaystyle (-4,2)}$
${\displaystyle (-4,2)}$
2. ${\displaystyle \left\{x:-{\tfrac {7}{3}}\leq x\leq -{\tfrac {1}{3}}\right\}}$
${\displaystyle \left[-{\tfrac {7}{3}},-{\tfrac {1}{3}}\right]}$
${\displaystyle \left[-{\tfrac {7}{3}},-{\tfrac {1}{3}}\right]}$
3. ${\displaystyle \{x:-\pi \leq x<\pi \}}$
${\displaystyle [-\pi ,\pi )}$
${\displaystyle [-\pi ,\pi )}$
4. ${\displaystyle \{x:x\leq {\tfrac {17}{9}}\}}$
${\displaystyle \left(-\infty ,{\tfrac {17}{9}}\right]}$
${\displaystyle \left(-\infty ,{\tfrac {17}{9}}\right]}$
5. ${\displaystyle \{x:5\leq x+1\leq 6\}}$
${\displaystyle [4,5]}$
${\displaystyle [4,5]}$
6. ${\displaystyle \left\{x:x-{\tfrac {1}{4}}<1\right\}}$
${\displaystyle \left(-\infty ,{\frac {5}{4}}\right)}$
${\displaystyle \left(-\infty ,{\frac {5}{4}}\right)}$
7. ${\displaystyle \{x:3>3x\}}$
${\displaystyle (-\infty ,1)}$
${\displaystyle (-\infty ,1)}$
8. ${\displaystyle \{x:0\leq 2x+1<3\}}$
${\displaystyle \left[-{\tfrac {1}{2}},1\right)}$
${\displaystyle \left[-{\tfrac {1}{2}},1\right)}$
9. ${\displaystyle \{x:5
${\displaystyle (5,6)}$
${\displaystyle (5,6)}$
10. ${\displaystyle \{x:5
${\displaystyle (-\infty ,\infty )}$
${\displaystyle (-\infty ,\infty )}$

### State the following intervals using set notation

11. ${\displaystyle [3,4]}$
${\displaystyle \{x:3\leq x\leq 4\}}$
${\displaystyle \{x:3\leq x\leq 4\}}$
12. ${\displaystyle [3,4)}$
${\displaystyle \{x:3\leq x<4\}}$
${\displaystyle \{x:3\leq x<4\}}$
13. ${\displaystyle (3,\infty )}$
${\displaystyle \{x:x>3\}}$
${\displaystyle \{x:x>3\}}$
14. ${\displaystyle \left(-{\tfrac {1}{3}},{\tfrac {1}{3}}\right)}$
${\displaystyle \left\{x:-{\tfrac {1}{3}}
${\displaystyle \left\{x:-{\tfrac {1}{3}}
15. ${\displaystyle \left(-\pi ,{\tfrac {15}{16}}\right)}$
${\displaystyle \left\{x:-\pi
${\displaystyle \left\{x:-\pi
16. ${\displaystyle (-\infty ,\infty )}$
${\displaystyle \{x:x\in \mathbb {R} \}}$
${\displaystyle \{x:x\in \mathbb {R} \}}$

### 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. ${\displaystyle |x+y|=|x|+|y|}$
false
false
18. ${\displaystyle |x+y|\geq |x|+|y|}$
false
false
19. ${\displaystyle |x+y|\leq |x|+|y|}$
true
true

### Evaluate the following expressions

20. ${\displaystyle 8^{\frac {1}{3}}}$
${\displaystyle 2}$
${\displaystyle 2}$
21. ${\displaystyle (-8)^{\frac {1}{3}}}$
${\displaystyle -2}$
${\displaystyle -2}$
22. ${\displaystyle \left({\frac {1}{8}}\right)^{\frac {1}{3}}}$
${\displaystyle {\frac {1}{2}}}$
${\displaystyle {\frac {1}{2}}}$
23. ${\displaystyle \left(8^{\frac {2}{3}}\right)\left(8^{\frac {3}{2}}\right)(8^{0})}$
${\displaystyle 8^{\frac {13}{6}}}$
${\displaystyle 8^{\frac {13}{6}}}$
24. ${\displaystyle \left(\left({\frac {1}{8}}\right)^{\frac {1}{3}}\right)^{7}}$
${\displaystyle {\frac {1}{128}}}$
${\displaystyle {\frac {1}{128}}}$
25. ${\displaystyle {\sqrt[{3}]{\frac {27}{8}}}}$
${\displaystyle {\frac {3}{2}}}$
${\displaystyle {\frac {3}{2}}}$
26. ${\displaystyle {\frac {4^{5}\cdot 4^{-2}}{4^{3}}}}$
${\displaystyle 1}$
${\displaystyle 1}$
27. ${\displaystyle \left({\sqrt {27}}\right)^{\frac {2}{3}}}$
${\displaystyle 3}$
${\displaystyle 3}$
28. ${\displaystyle {\frac {\sqrt {27}}{\sqrt[{3}]{9}}}}$
${\displaystyle 3^{\frac {5}{6}}}$
${\displaystyle 3^{\frac {5}{6}}}$

### Simplify the following

29. ${\displaystyle x^{3}+3x^{3}}$
${\displaystyle 4x^{3}}$
${\displaystyle 4x^{3}}$
30. ${\displaystyle {\frac {x^{3}+3x^{3}}{x^{2}}}}$
${\displaystyle 4x}$
${\displaystyle 4x}$
31. ${\displaystyle (x^{3}+3x^{3})^{3}}$
${\displaystyle 64x^{9}}$
${\displaystyle 64x^{9}}$
32. ${\displaystyle {\frac {x^{15}+x^{3}}{x}}}$
${\displaystyle x^{14}+x^{2}}$
${\displaystyle x^{14}+x^{2}}$
33. ${\displaystyle (2x^{2})(3x^{-2})}$
${\displaystyle 6}$
${\displaystyle 6}$
34. ${\displaystyle {\frac {x^{2}y^{-3}}{x^{3}y^{2}}}}$
${\displaystyle {\frac {1}{xy^{5}}}}$
${\displaystyle {\frac {1}{xy^{5}}}}$
35. ${\displaystyle {\sqrt {x^{2}y^{4}}}}$
${\displaystyle |xy^{2}|}$
${\displaystyle |xy^{2}|}$
36. ${\displaystyle \left({\frac {8x^{6}}{y^{4}}}\right)^{\frac {1}{3}}}$
${\displaystyle {\frac {2x^{2}}{y^{\frac {4}{3}}}}}$
${\displaystyle {\frac {2x^{2}}{y^{\frac {4}{3}}}}}$

### Find the roots of the following polynomials

37. ${\displaystyle x^{2}-1}$
${\displaystyle x=\pm 1}$
${\displaystyle x=\pm 1}$
38. ${\displaystyle x^{2}+2x+1}$
${\displaystyle x=-1}$
${\displaystyle x=-1}$
39. ${\displaystyle x^{2}+7x+12}$
${\displaystyle x=-3,x=-4}$
${\displaystyle x=-3,x=-4}$
40. ${\displaystyle 3x^{2}-5x-2}$
${\displaystyle x=2,x=-{\frac {1}{3}}}$
${\displaystyle x=2,x=-{\frac {1}{3}}}$
41. ${\displaystyle x^{2}+{\frac {5}{6}}x+{\frac {1}{6}}}$
${\displaystyle x=-{\frac {1}{3}},x=-{\frac {1}{2}}}$
${\displaystyle x=-{\frac {1}{3}},x=-{\frac {1}{2}}}$
42. ${\displaystyle 4x^{3}+4x^{2}+x}$
${\displaystyle x=0,x=-{\frac {1}{2}}}$
${\displaystyle x=0,x=-{\frac {1}{2}}}$
43. ${\displaystyle x^{4}-1}$
${\displaystyle x=\pm i,x=\pm 1}$
${\displaystyle x=\pm i,x=\pm 1}$
44. ${\displaystyle x^{3}+2x^{2}-4x-8}$
${\displaystyle x=\pm 2}$
${\displaystyle x=\pm 2}$

### Factor the following expressions

45. ${\displaystyle 4a^{2}-ab-3b^{2}}$
${\displaystyle (4a+3b)(a-b)}$
${\displaystyle (4a+3b)(a-b)}$
46. ${\displaystyle (c+d)^{2}-4}$
${\displaystyle (c+d+2)(c+d-2)}$
${\displaystyle (c+d+2)(c+d-2)}$
47. ${\displaystyle 4x^{2}-9y^{2}}$
${\displaystyle (2x+3y)(2x-3y)}$
${\displaystyle (2x+3y)(2x-3y)}$

### Simplify the following

48. ${\displaystyle {\frac {x^{2}-1}{x+1}}}$
${\displaystyle x-1,x\neq -1}$
${\displaystyle x-1,x\neq -1}$
49. ${\displaystyle {\frac {3x^{2}+4x+1}{x+1}}}$
${\displaystyle 3x+1,x\neq -1}$
${\displaystyle 3x+1,x\neq -1}$
50. ${\displaystyle {\frac {4x^{2}-9}{4x^{2}+12x+9}}}$
${\displaystyle {\frac {2x-3}{2x+3}}}$
${\displaystyle {\frac {2x-3}{2x+3}}}$
51. ${\displaystyle {\frac {x^{2}+y^{2}+2xy}{x(x+y)}}}$
${\displaystyle {\frac {x+y}{x}},x\neq -y}$
${\displaystyle {\frac {x+y}{x}},x\neq -y}$

## Functions

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

a. Compute ${\displaystyle f(0)}$ , ${\displaystyle f(2)}$ , and ${\displaystyle f(-1.2)}$ .
${\displaystyle f(0)=0}$ , ${\displaystyle f(2)=4}$ , and ${\displaystyle f(-1.2)=1.44}$
${\displaystyle f(0)=0}$ , ${\displaystyle f(2)=4}$ , and ${\displaystyle f(-1.2)=1.44}$
b. What are the domain and range of ${\displaystyle f}$ ?
Domain is ${\displaystyle (-\infty ,\infty )}$ ; range is ${\displaystyle [0,\infty )}$
Domain is ${\displaystyle (-\infty ,\infty )}$ ; range is ${\displaystyle [0,\infty )}$
c. Does ${\displaystyle f}$ have an inverse? If so, find a formula for it.
No, ${\displaystyle f}$ is not one-to-one. For example, both ${\displaystyle x=1}$ and ${\displaystyle x=-1}$ result in ${\displaystyle f(x)=1}$ .
No, ${\displaystyle f}$ is not one-to-one. For example, both ${\displaystyle x=1}$ and ${\displaystyle x=-1}$ result in ${\displaystyle f(x)=1}$ .

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

a. Give formulae for
i. ${\displaystyle f+g}$
${\displaystyle (f+g)(x)=x+2+{\frac {1}{x}}}$
${\displaystyle (f+g)(x)=x+2+{\frac {1}{x}}}$
ii. ${\displaystyle f-g}$
${\displaystyle (f-g)(x)=x+2-{\frac {1}{x}}}$
${\displaystyle (f-g)(x)=x+2-{\frac {1}{x}}}$
iii. ${\displaystyle g-f}$
${\displaystyle (g-f)(x)={\frac {1}{x}}-x-2}$
${\displaystyle (g-f)(x)={\frac {1}{x}}-x-2}$
iv. ${\displaystyle f\times g}$
${\displaystyle (f\times g)(x)=1+{\frac {2}{x}}}$
${\displaystyle (f\times g)(x)=1+{\frac {2}{x}}}$
v. ${\displaystyle {\frac {f}{g}}}$
${\displaystyle \left({\frac {f}{g}}\right)(x)=x^{2}+2x}$
${\displaystyle \left({\frac {f}{g}}\right)(x)=x^{2}+2x}$
vi. ${\displaystyle {\frac {g}{f}}}$
${\displaystyle \left({\frac {g}{f}}\right)(x)={\frac {1}{x^{2}+2x}}}$
${\displaystyle \left({\frac {g}{f}}\right)(x)={\frac {1}{x^{2}+2x}}}$
vii. ${\displaystyle f\circ g}$
${\displaystyle (f\circ g)(x)={\frac {1}{x}}+2}$
${\displaystyle (f\circ g)(x)={\frac {1}{x}}+2}$
viii. ${\displaystyle g\circ f}$
${\displaystyle (g\circ f)(x)={\frac {1}{x+2}}}$
${\displaystyle (g\circ f)(x)={\frac {1}{x+2}}}$
b. Compute ${\displaystyle f(g(2))}$ and ${\displaystyle g(f(2))}$ .
${\displaystyle f(g(2))=2.5\ ,\ g(f(2))=0.25}$
${\displaystyle f(g(2))=2.5\ ,\ g(f(2))=0.25}$
c. Do ${\displaystyle f}$ and ${\displaystyle g}$ have inverses? If so, find formulae for them.
${\displaystyle f^{-1}(x)=x-2\ ,\ g^{-1}(x)={\frac {1}{x}}}$
${\displaystyle f^{-1}(x)=x-2\ ,\ g^{-1}(x)={\frac {1}{x}}}$
54. Does this graph represent a function?
Yes.
Yes.

55. Consider the following function

${\displaystyle f(x)={\begin{cases}-{\frac {1}{9}}&{\mbox{if }}x<-1\\2&{\mbox{if }}-1\leq x\leq 0\\x+3&{\mbox{if }}x>0\end{cases}}}$
a. What is the domain?
${\displaystyle (-\infty ,\infty )}$
${\displaystyle (-\infty ,\infty )}$
b. What is the range?
${\displaystyle \left(-{\tfrac {1}{9}},\infty \right)}$
${\displaystyle \left(-{\tfrac {1}{9}},\infty \right)}$
c. Where is ${\displaystyle f}$ continuous?
${\displaystyle x>0}$
${\displaystyle x>0}$

56. Consider the following function

${\displaystyle f(x)={\begin{cases}x^{2}&{\mbox{if }}x>0\\-1&{\mbox{if }}x\leq 0\end{cases}}}$
a. What is the domain?
${\displaystyle (-\infty ,\infty )}$
${\displaystyle (-\infty ,\infty )}$
b. What is the range?
${\displaystyle (-1,\infty )}$
${\displaystyle (-1,\infty )}$
c. Where is ${\displaystyle f}$ continuous?
${\displaystyle x>0}$
${\displaystyle x>0}$

57. Consider the following function

${\displaystyle f(x)={\frac {\sqrt {2x-3}}{x-10}}}$
a. What is the domain?
${\displaystyle (1.5,10)\cup (10,\infty )}$
${\displaystyle (1.5,10)\cup (10,\infty )}$
b. What is the range?
${\displaystyle (-\infty ,\infty )}$
${\displaystyle (-\infty ,\infty )}$
c. Where is ${\displaystyle f}$ continuous?
${\displaystyle (1.5,10)\cup (10,\infty )}$
${\displaystyle (1.5,10)\cup (10,\infty )}$

58. Consider the following function

${\displaystyle f(x)={\frac {x-7}{x^{2}-49}}}$
a. What is the domain?
${\displaystyle (-\infty ,-7)\cup (-7,\infty )}$
${\displaystyle (-\infty ,-7)\cup (-7,\infty )}$
b. What is the range?
${\displaystyle (-\infty ,\infty )}$
${\displaystyle (-\infty ,\infty )}$
c. Where is ${\displaystyle f}$ continuous?
${\displaystyle (-\infty ,-7)\cup (-7,7)\cup (7,\infty )}$
${\displaystyle (-\infty ,-7)\cup (-7,7)\cup (7,\infty )}$

## Graphing

59. Find the equation of the line that passes through the point (1,-1) and has slope 3.
${\displaystyle 3x-y=4}$
${\displaystyle 3x-y=4}$
60. Find the equation of the line that passes through the origin and the point (2,3).
${\displaystyle 3x-2y=0}$
${\displaystyle 3x-2y=0}$
 ← Graphing linear functions Calculus Limits → Precalculus/Exercises