Question 1

Which of the following is the correct definition of a piecewise function?

Accepted Answer

A function that is defined by two or more equations over a specified domain.

Question 2

Given the function $$f(x) = \begin{cases} 3x + 5 & 	ext{if } -4 \leq x \u003c 1 \ 4x - 2 & 	ext{if } x \geq 1 \end{cases}$$, what is $$f(-2)$$?

Accepted Answer

$$f(-2) = 3(-2) + 5 = -1$$

Question 3

Which of the following is the parent function for the absolute value?

Accepted Answer

$$y = |x|$$

Question 4

What is the domain of the function $$f(x) = \sqrt{x}$$?

Accepted Answer

$$[0, \infty)$$

Question 5

Given $$f(x) = \frac{1}{x}$$, which of the following statements is true about its domain?

Accepted Answer

The domain is all real numbers except $$x = 0.$$

Question 6

Evaluate $$f(0)$$ for the piecewise function $$f(x) = \begin{cases} 3x + 1 & x \u003c 1 \ -2x + 1 & x \geq 1 \end{cases}.$$

Accepted Answer

$$f(0) = 3(0) + 1 = 1$$

Question 7

Which of the following functions is NOT a parent function listed in the library of functions?

Accepted Answer

$$y = x^4$

Question 8

A function $$f is $$defined as $$f(x) = \begin{cases} 2x & x \u003c 0 \ x^2$ & x \geq 0 \end{cases}. $$What is $$f(-3) + f(2)$$?

Accepted Answer

$$f(-3) = 2(-3) = -6$$, $$f(2) = 2^2$ = 4$$, so $$-6 + 4 = -2$$

Question 9

Which of the following best describes the graph of $$y = |x|$$?

Accepted Answer

A V-shaped graph opening upwards.

Question 10

For the function $$f(x) = \begin{cases} 3x + 5 & -4 \leq x \u003c 1 \ 4x - 2 & x \geq 1 \end{cases}$$, what is the value of $$f(1)$$?

Accepted Answer

$$f(1) = 4(1) - 2 = 2$$

Question 11

Which of the following is the range of the quadratic parent function $$y = x^2$?

Accepted Answer

$$[0, \infty)$$

Question 12

A medical dosage is modeled by the function $$f(x) = \begin{cases} 2x + 10 & x \u003c 5 \ 3x - 5 & x \geq 5 \end{cases}$$, where $$x is $$the number of hours after administration. What is the dosage at $$x = 6$$ hours?

Accepted Answer

$$f(6) = 3(6) - 5 = 13$$

Question 13

Which of the following is true about the function $$y = x^3$?

Accepted Answer

It is an odd function and its graph is symmetric about the origin.

Question 14

A piecewise function is given by $$f(x) = \begin{cases} x^2$ & x \u003c 2 \ 2x + 1 & x \geq 2 \end{cases}. $$What is $$f(2)$$?

Accepted Answer

$$f(2) = 2(2) + 1 = 5$$

Question 15

Which function from the library of functions has a domain of all real numbers except $$x = 0$$ and a range of all real numbers except $$y = 0$$?

Accepted Answer

$$y = \frac{1}{x}$$

Question 16

If $$f(x) = |x|$$ and $$g(x) = x^2$, what is the difference quotient $$\frac{f(x+h) - f(x)}{h}$$ when x = -2$ and h = 1$?

Accepted Answer

$$\frac{|(-2)+1| - |-2|}{1} = \frac{|-1| - 2}{1} = -1$$

Question 17

Which of the following is the correct domain for the piecewise function $$f(x) = \begin{cases} 3x + 5 & -4 \leq x \u003c 1 \ 4x - 2 & x \geq 1 \end{cases}$$?

Accepted Answer

$$[-4, \infty)$$

Question 18

A chemical reaction rate is modeled by $$f(x) = \sqrt{x}$$, where $$x is $$the concentration in mol/L. If the concentration is $$9$$ mol/L, what is the reaction rate?

Accepted Answer

$$f(9) = \sqrt{9} = 3$$

Library of Functions and Piecewise-defined Functions: Study Notes

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