Question 1

Given the function $$f(x) = 2x^3$ + 6x^2$ - 18x + 8$$, use calculus or graphing techniques to determine the $$x-$$value at which $$f$$ attains its relative minimum, and find the value of this minimum.

Accepted Answer

$$x = 1,\ f(1) = -2$$

Question 2

A function $$f is $$defined by the piecewise rule:

$$\begin{cases} 3x + 5 & 	ext{if } x \u003c 0 \ 2x + 7 & 	ext{if } x \geq 0 \end{cases}$$

What is $$f(-1)$$?

Accepted Answer

$$2$$

Question 3

Given the graph of a function $$f$$ with domain $$(-\infty, 6]$$ and range $$(-\infty, 3]$$, which of the following statements is true?

Accepted Answer

The function never attains a value greater than $$3.$$

Question 4

For the function $$f(x) = x^2$ - 4x + 3$$, what is the simplified form of the difference quotient $$\dfrac{f(x+h) - f(x)}{h}$$?

Accepted Answer

$$2x + h - 4$$

Question 5

A function $$f is $$constant on the interval $$(-\infty, -2)$$ and decreasing on $$(-2, \infty). $$Which of the following best describes the behavior of $$f$$?

Accepted Answer

The function is constant for $$x  -2.$$

Question 6

Suppose a function $$f$$ has a relative maximum at $$x=3$$ with $$f(3)=5$$, and relative minima at $$x=2$$ and $$x=4$$ with $$f(2)=1$$ and $$f(4)=1. $$Which of the following is true?

Accepted Answer

The function has relative minima at $$x=2$$ and $$x=4$$, both with value $$1.$$

Question 7

Given the function $$f(x) = -5x + 8$$, what is the value of the difference quotient $$\dfrac{f(x+h) - f(x)}{h}$$ for $$h 
eq 0$$?

Accepted Answer

$$-5$$

Question 8

A function $$f is $$defined on $$[25,75]$$ and its range is $$[32,38]. $$Which of the following best describes the possible real-world context for $$f$$?

Accepted Answer

The percent body fat in adult women as a function of age.

Question 9

Given the function $$f(x) = 6$$, what is the value of the difference quotient $$\dfrac{f(x+h) - f(x)}{h}$$ for $$h 
eq 0$$?

Accepted Answer

$$6$$

Question 10

A function $$f is $$defined by $$f(x) = x + 2$$ for $$x \u003c 4$$ and $$f(x) = x - 2$$ for $$x \geq 4. $$What is the range of $$f$$?

Accepted Answer

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

Question 11

Suppose the graph of a function $$f is $$symmetric with respect to the $$y-$$axis. Which of the following statements is true?

Accepted Answer

The function is even.

Question 12

Given a function $$f$$ with domain $$(-\infty, \infty)$$ and range $$[-4, \infty)$$, what is the minimum value of $$f$$?

Accepted Answer

$$-4$$

Question 13

If a function $$f is $$increasing on $$(0, \infty)$$ and decreasing on $$(-\infty, 0)$$, what can you conclude about $$x=0$$?

Accepted Answer

$$x=0 is a $$relative minimum.

Question 14

Given a function $$f$$ with $$x-$$intercepts at $$x=3$$ and $$x=7$$, and $$y-$$intercept at $$y=21$$, which of the following is true?

Accepted Answer

The function crosses the $$x-$$axis at $$x=3$$ and $$x=7.$$

Question 15

Suppose $$f(x) is $$defined for all real $$x$$ and is odd. Which of the following must be true for all $$x in $$the domain?

Accepted Answer

$$f(-x) = -f(x)$$

Question 16

A function $$f is $$increasing on $$(5, \infty)$$, decreasing on $$(0,5)$$, and constant on $$(-\infty,0). $$What is the $$x-$$value at which $$f$$ attains its relative minimum?

Accepted Answer

$$x=5$$

Question 17

Given the function $$f(x) = 2x^3$ + 6x^2$ - 18x + 8$$, what is the $$x-$$value at which $$f$$ attains its relative maximum?

Accepted Answer

$$x=-3$$

Question 18

If $$f(x) is $$defined on $$(-\infty,6]$$ and $$f(5.5) = 1$$, what is the value of $$f(5.5)$$?

Accepted Answer

$$1$$

Question 19

For the function $$f(x) = 2x^3$ + 6x^2$ - 18x + 8$$, what is the value of $$f(-3)$$?

Accepted Answer

$$62$$

Question 20

Given a function $$f$$ with domain $$(-\infty, \infty)$$ and range $$(-\infty, \infty)$$, which of the following could be the range of $$f(x) = x + 2$$ for $$x \u003c 4$$ and $$f(x) = x - 2$$ for $$x \geq 4$$?

Accepted Answer

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

College Algebra Study Guide: Functions, Graphs, and Difference Quotients

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