# Calculus/Precalculus/Exercises

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

## Algebra

### Convert to interval notation

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

### State the following intervals using set notation

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

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

### Evaluate the following expressions

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

### Simplify the following

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

### Find the roots of the following polynomials

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

### Factor the following expressions

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

### Simplify the following

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

## Functions

52. Let $f(x)=x^{2}$  .

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

53. Let $f(x)=x+2$  , $g(x)=1/x$  .

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

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?
$(-\infty ,\infty )$
$(-\infty ,\infty )$
b. What is the range?
$\left(-{\tfrac {1}{9}},\infty \right)$
$\left(-{\tfrac {1}{9}},\infty \right)$
c. Where is $f$  continuous?
$x>0$
$x>0$

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?
$(-\infty ,\infty )$
$(-\infty ,\infty )$
b. What is the range?
$(-1,\infty )$
$(-1,\infty )$
c. Where is $f$  continuous?
$x>0$
$x>0$

57. Consider the following function

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

58. Consider the following function

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

## Graphing

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