Q: A household uses 712 kilowatt-hours of electricity in a month. The monthly charge $$c is $$given by $$c = 18 + 0.128x$$, where $$x is $$the number of kilowatt-hours. What is the monthly charge?

$$c = 18 + 0.128 \times 712 = 109.54$$

Question 1

Which of the following relations is a function?

$$\{(1, 4), (2, 5), (3, 6), (4, 7)\}$$

Accepted Answer

Yes, because each first component is paired with exactly one second component.

Question 2

Given the relation $$\{(-3, 9), (-2, 4), (0, 0), (1, 1), (-3, 8)\}$$, is this a function?

Accepted Answer

No, because the first component $$-3 is $$paired with two different second components.

Question 3

Which of the following best describes the domain of the function $$f(x) = \sqrt{12 - 3x}$$?

Accepted Answer

$$(-\infty, 4]$$

Question 4

Let $$f(x) = -x^2$ + 3x. $$Find the difference quotient $$\dfrac{f(1+h) - f(1)}{h}$$ for $$h 
eq 0.$$

Accepted Answer

$$-2h - h^2$ + 3$$

Question 5

Given $$f(x) = \dfrac{x^2$ - 1}{x + 2}$$ and $$g(x) = \dfrac{1 + x}{x^2$ - 4}$$, find $$\left(\dfrac{f}{g}\right)(x).$$

Accepted Answer

$$\dfrac{(x^2$ - 1)(x^2$ - 4)}{(x + 2)(1 + x)}$$

Question 6

Which of the following equations does NOT define $$y as a $$function of $$x$$?

Accepted Answer

$$x^2$$ - $$4y^2 = 1$

Question 7

Let $$f(x) = \sqrt{x + 3}. $$Find the difference quotient $$\dfrac{f(x+h) - f(x)}{h}.$$

Accepted Answer

$$\dfrac{\sqrt{x + h + 3} - \sqrt{x + 3}}{h}$$

Question 8

Suppose $$f(x) = x^2$ + 5x. $$What is the domain of $$f$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 9

Let $$f(x) = \dfrac{x - 1}{|x| - 1}. $$Which value is NOT in the domain of $$f$$?

Accepted Answer

$$x = 1$$

Question 10

If $$f(x) = -x^2$ - 7x$$, find $$f(3).$$

Accepted Answer

$$-9 - 21 = -30$$

Question 11

Let $$f(x) = \dfrac{x - B}{x - A}$$, $$f(2) = 0$$ and $$f(1) is $$undefined. What are the values of $$A$$ and $$B$$?

Accepted Answer

$$A = 1, B = 2$$

Question 12

Given $$f(x) = -x^2$ + 2x$$ and $$g(x) = x + 5$$, find $$(f - g)(x)$$ and its domain.

Accepted Answer

$$(f - g)(x) = -x^2$ + x - 5$$, domain: $$(-\infty, \infty)$$

Question 13

A household uses 712 kilowatt-hours of electricity in a month. The monthly charge $$c is $$given by $$c = 18 + 0.128x$$, where $$x is $$the number of kilowatt-hours. What is the monthly charge?

Accepted Answer

$$c = 18 + 0.128 	imes 712 = 109.54$$

Question 14

Which of the following best describes the range of a function?

Accepted Answer

The set of all possible output values.

Question 15

Given $$f(x) = \dfrac{3x + 8}{2x - A}$$ and $$f(0) = 2$$, what is the value of $$A$$?

Accepted Answer

$$A = -4$$

Question 16

Let $$f(x) = \sqrt{x - 1}$$ and $$g(x) = \sqrt{2 - x - 1}. $$What is the domain of $$f(x) + g(x)$$?

Accepted Answer

$$[1, 2]$$

Question 17

If $$f(x) = mx + b$$, show that $$\dfrac{f(x + h) - f(x)}{h} = m.$$

Accepted Answer

Because $$f(x + h) - f(x) = m(x + h) + b - (mx + b) = mh$$, so $$\dfrac{mh}{h} = m.$$

Question 18

Given $$f(x) = \dfrac{1}{x}$$ and $$\left(\dfrac{f}{g}\right)(x) = \dfrac{x + 1}{x^2$ - x}$$, find $$g(x).$$

Accepted Answer

$$g(x) = \dfrac{x^2$ - x}{x + 1}$$

Functions and Their Graphs: Foundations and Operations

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