Question 1

Which of the following statements is TRUE about even functions?

Accepted Answer

For every $$x in $$the domain, $$f(-x) = f(x).$$

Question 2

Given the function $$f(x) = x^3$ - x$$, determine whether it is even, odd, or neither.

Accepted Answer

Odd

Question 3

A function $$f is $$said to be decreasing on an interval $$I if$$:

Accepted Answer

For any $$x_1  f$$($$x_2$$)$$

Question 4

Suppose the graph of $$f$$ has a local maximum at $$x = 2. $$Which of the following must be true?

Accepted Answer

There is an open interval around $$x = 2$$ where $$f(2) \geq f(x)$$ for all $$x in $$that interval.

Question 5

If $$f(x) is $$defined on $$[-3, 5]$$ and is continuous, what does the Extreme Value Theorem guarantee?

Accepted Answer

That $$f(x)$$ has an absolute maximum and an absolute minimum on $$[-3, 5].$$

Question 6

Given $$f(x) = x^2$ + 1$$, what is the average rate of change from $$x = 1 to$$ $$x = 2$$?

Accepted Answer

$$3$$

Question 7

Which of the following best describes the secant line to the graph of $$f$$ through points $$(a, f(a))$$ and $$(b, f(b))$$?

Accepted Answer

It is the line passing through $$(a, f(a))$$ and $$(b, f(b)).$$

Question 8

Suppose $$f(x) is $$constant on the interval $$[1, 3]. $$What is the average rate of change of $$f$$ from $$x = 1 to$$ $$x = 3$$?

Accepted Answer

$$0$$

Question 9

If $$f(x) is $$odd, which of the following must be true for all $$x in $$the domain of $$f$$?

Accepted Answer

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

Question 10

A company’s profit function $$P(t) (in $$thousands of dollars) over time $$t (in $$years) is given by $$P(t) = -2t^2$ + 8t + 10. $$What is the average rate of change of profit from year 1 to year 3?

Accepted Answer

$$-2$$

Question 11

Given the function $$f(x) = |x|$$, which of the following is true?

Accepted Answer

It is an even function.

Question 12

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

Accepted Answer

$$x = 0 is a $$local maximum.

Question 13

Suppose $$f(x) is $$defined on $$[0, 4]$$ and has local maxima at $$x = 1$$ and $$x = 3$$, and a local minimum at $$x = 2. If$$ $$f(0) = 2$$, $$f(1) = 5$$, $$f(2) = 1$$, $$f(3) = 4$$, and $$f(4) = 3$$, what is the absolute maximum value of $$f on$$ $$[0, 4]$$?

Accepted Answer

$$5$$

Question 14

Which of the following is NOT a property of an even function?

Accepted Answer

Its graph is symmetric with respect to the origin.

Question 15

A function $$f$$ has a local minimum at $$x = 2$$ and a local maximum at $$x = 5. $$Which of the following could be true about the intervals of increase and decrease?

Accepted Answer

$$f is $$increasing on $$(2, 5)$$ and decreasing on $$(-\infty, 2)$$ and $$(5, \infty).$$

Question 16

If $$f(x) = x^4$ - 2x^2$ + 1$$, what is the average rate of change from $$x = -1 to$$ $$x = 2$$?

Accepted Answer

$$-3$$

Question 17

Which of the following real-world scenarios best illustrates the concept of a local maximum?

Accepted Answer

The highest temperature reached during a single day.

Question 18

Given $$f(x) = x^2$ - 4x + 3$$, find the equation of the secant line through $$x = 1$$ and $$x = 3.$$

Accepted Answer

$$y = 2x - 3$$

Question 19

A function $$f is $$defined on $$[a, b]. If$$ $$f(a) = f(b)$$ and $$f(x) \u003c f(a)$$ for all $$x in$$ $$(a, b)$$, what can you conclude?

Accepted Answer

$$f$$ has an absolute maximum at $$x = a$$ and $$x = b.$$

Properties of Functions: Even/Odd Functions, Intervals, Extrema, and Average Rate of Change

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