Its graph is symmetric with respect to the y-axis.

$f(1)$ is the smallest value of $f$ near $x = 1$, and $f(4)$ is the largest value of $f$ near $x = 4$.

Question 1

Which of the following statements correctly describes an even function?

Accepted Answer

A function $f$ is even if $f(-x) = f(x)$ for all $x$ in its domain.

Question 2

Given the function $h(x) = x^3 + 4x$, determine whether it is even, odd, or neither.

Accepted Answer

Odd

Question 3

Which of the following functions is symmetric with respect to the y-axis?

Accepted Answer

$g(x) = x^2$

Question 4

A function $f$ is defined on the interval $[a, b]$ and is continuous. According to the Extreme Value Theorem, which statement is true?

Accepted Answer

The function $f$ must have an absolute maximum and absolute minimum on $[a, b]$.

Question 5

For $f(x) = x^2 + 1$, what is the average rate of change from $x = 0$ to $x = 2$?

Accepted Answer

$2$

Question 6

If the graph of a function is symmetric with respect to the origin, what can you conclude about the function?

Accepted Answer

It is an odd function.

Question 7

Suppose $f(x)$ is decreasing on the interval $[1, 3]$. Which of the following must be true for any $x_1 < x_2$ in $[1, 3]$?

Accepted Answer

$f(x_1) > f(x_2)$

Question 8

Given $g(x) = x^3 + 3$, determine if $g$ is even, odd, or neither.

Accepted Answer

Neither

Question 9

A function $f$ has a local maximum at $x = 2$ and a local minimum at $x = 5$. Which statement is true?

Accepted Answer

$f(2)$ is the largest value of $f$ near $x = 2$, and $f(5)$ is the smallest value of $f$ near $x = 5$.

Question 10

Which of the following is the correct formula for the average rate of change of a function $f$ from $x = a$ to $x = b$?

Accepted Answer

$\frac{f(b) - f(a)}{b - a}$

Question 11

If $F(x) = 1 + |x|$, is $F$ even, odd, or neither?

Accepted Answer

Even

Question 12

A function $f$ is constant on the interval $[2, 5]$. What does this mean about $f(x)$ for $x$ in $[2, 5]$?

Accepted Answer

$f(x)$ is the same for all $x$ in $[2, 5]$.

Question 13

Suppose $f(x)$ is defined on $[0, 5]$ and $f(3) = 6$ is the largest value of $f$ on this interval. What is the absolute maximum of $f$?

Accepted Answer

$6$

Question 14

Which of the following real-world scenarios best illustrates the concept of average rate of change?

Accepted Answer

Finding the difference in temperature between two times and dividing by the time elapsed.

Question 15

Given $g(x) = x^2 - 3x + 2$, find the average rate of change from $x = 1$ to $x = 3$.

Accepted Answer

$1$

Question 16

If a function $f$ is increasing on $[a, b]$, which of the following is true for any $x_1 < x_2$ in $[a, b]$?

Accepted Answer

$f(x_1) < f(x_2)$

Question 17

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

Accepted Answer

Its graph is symmetric with respect to the y-axis.

Question 18

A secant line passes through the points $(a, f(a))$ and $(b, f(b))$ on the graph of $f$. What is the slope of this secant line?

Accepted Answer

$\frac{f(b) - f(a)}{b - a}$

Question 19

Suppose a function $f$ has a local minimum at $x = 1$ and a local maximum at $x = 4$. Which of the following is true?

Accepted Answer

$f(1)$ is the smallest value of $f$ near $x = 1$, and $f(4)$ is the largest value of $f$ near $x = 4$.

Properties of Functions: Even and Odd Functions, Increasing/Decreasing Intervals, and Extrema

Study Guide - Practice Questions