Calculus/Precalculus/Solutions

Convert to interval notation

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

$\mathbf{(-4,2)}$

2. $\{x:-\frac{7}{3} \leq x \leq -\frac{1}{3}\}$

$\mathbf{[-\frac{7}{3},-\frac{1}{3}]}$

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

$\mathbf{[-\pi,\pi)}$

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

$\mathbf{(-\infty, \frac{17}{9}]}$

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

$4\leq x\leq5$
$\mathbf{[4, 5]}$

6. $\{x:x - 1/4 < 1\} \,$

$x<1 \frac{1}{4}=\frac{5}{4}$
$\mathbf{(-\infty, \frac{5}{4})}$

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

1 > x
x < 1
$\mathbf{(-\infty, 1)}$

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

$-1\leq 2x\leq 2$
$-\frac{1}{2}\leq x<1$
$\mathbf{[-\frac{1}{2}, 1)}$

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

This is equivalent to 5 < x < 6
$\mathbf{(5,6)}$

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

It helps to draw a picture to determine the set of numbers described:

A number in the set can be on either the red or blue line, so the entire number line is included.
$\mathbf{(-\infty,\infty)}$

State the following intervals using set notation

11. $[3,4] \,$

$\mathbf{\{x:3\leq x\leq 4\}}$

12. $[3,4) \,$

$\mathbf{\{x:3\leq x<4\}}$

13. $(3,\infty)$

$\mathbf{\{x:x>3\}}$

14. $(-\frac{1}{3}, \frac{1}{3}) \,$

$\mathbf{\{x:-\frac{1}{3}<x<\frac{1}{3}\}}$

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

$\mathbf{\{x:-\pi<x<\frac{15}{16}\}}$

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

$\mathbf{\{x:x\in\Re\}}$

Which one of the following is a true statement?

17. $|x+y| = |x| + |y| \,$

Let x = − 5,y = 5. Then
| x + y | = | − 5 + 5 | = | 0 | = 0, and
| x | + | y | = | − 5 | + | 5 | = 5 + 5 = 10
Thus, $|x+y| \neq |x| + |y|$
false

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

Using the same example as above, we have $|x+y|\ngeq |x| + |y|$.
false

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

true

Evaluate the following expressions

20. $8^{1/3} \,$

$(2^3)^{1/3}=2^1=\mathbf{2}$

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

$(-2^3)^{1/3}=-2^1=\mathbf{-2}$

22. $\bigg(\frac{1}{8}\bigg)^{1/3} \,$

$(\frac{1}{2^3})^{1/3}=(2^{-3})^{1/3}=2^{-1}=\mathbf{\frac{1}{2}}$

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

$8^{\frac{2}{3}+\frac{3}{2}+0}=8^{\frac{4}{6}+\frac{8}{6}}=8^{\frac{13}{6}}=(2^3)^{\frac{13}{6}}=\mathbf{2^{13/2}}$

24. $\bigg( \bigg(\frac{1}{8}\bigg)^{1/3} \bigg)^7$

$((\frac{1}{2^3})^{1/3})^7=((2^{-3})^{1/3})^7=(2^{-1})^7=2^{-7}=\frac{1}{2^7}=\mathbf{\frac{1}{128}}$

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

$(\frac{27}{8})^{1/3}=(\frac{3^3}{2^3})^{1/3}=\frac{3^1}{2^1}=\mathbf{\frac{3}{2}}$

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

$4^{5-2-3}=4^0=\mathbf{1}$

27. $\bigg(\sqrt{27}\bigg)^{2/3}$

$((3^3)^{1/2})^{2/3}=(3^\frac{3}{2})^\frac{2}{3}=3^1=\mathbf{3}$

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

$\frac{(3^3)^{1/2}}{(3^2)^{1/3}}=\frac{3^\frac{3}{2}}{3^\frac{2}{3}}=3^{\frac{3}{2}-\frac{2}{3}}=3^{\frac{9}{6}-\frac{4}{6}}=\mathbf{3^{5/6}}$

Simplify the following

29. $x^3 + 3x^3 \,$

$\mathbf{ 4x^3 }$

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

$\mathbf{ 4x }$

31. $(x^3+3x^3)^3 \,$

$\mathbf{ 64x^9 }$

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

$\mathbf{ x^{14} + x^2 }$

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

$\mathbf{ 6 }$

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

$\mathbf{ \frac{1}{xy^5} }$

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

$\mathbf{ xy^2 }$

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

$\mathbf{ \frac{2x^2}{y^{\frac{4}{3}}}}$

Functions

52. Let f(x) = x².

a. Compute f(0) and f(2).

f(0) = 0, f(2) = 4

b. What are the domain and range of f?

The domain is $(-\infty,\infty)$; the range is $[0,\infty)$,

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

No, since f isn't one-to-one; for example, f( − 1) = f(1) = 1.

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

a. Give formulae for

i. f + g

(f + g)(x) = x + 2 + 1 / x = (x² + 2x + 1) / x.

ii. f − g

(f − g)(x) = x + 2 − 1 / x = (x² + 2x − 1) / x.

iii. g − f

(g − f)(x) = 1 / x − x − 2 = (1 − x² − 2x) / x.

iv. $f\times g$

$(f\times g)(x)=(x+2)/x$.

v. f / g

(f / g)(x) = x(x + 2) provided $x\ne0$. Note that 0 is not in the domain of f / g, since it's not in the domain of g, and you can't divide by something that doesn't exist!

vi. g / f

(g / f)(x) = 1 / [x(x + 2)]. Although 0 is still not in the domain, we don't need to state it now, since 0 isn't in the domain of the expression 1 / [x(x + 2)] either.

vii. $f\circ g$

$(f\circ g)(x)=1/x+2=(2x+1)/x$.

viii. $g\circ f$

$(g\circ f)(x)=1/(x+2)$.

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.

Yes; f ^{− 1}(x) = x − 2 and g ^{− 1}(x) = 1 / x. Note that g and its inverse are the same.

54. Does this graph represent a function?

As pictured, by the Vertical Line test, this graph represents a function.

55. Consider the following function

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

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

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

a. What is the domain?

b. What is the range?

c. Where is f continuous?

58. Consider the following function

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

a. What is the domain?

b. What is the range?

c. Where is f continuous?