Q: A function $f$ is constant on an interval if:

For any $x_1 < x_2$, $f(x_1) = f(x_2)$.

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

Odd, because $f(-x) = -f(x)$ for all $x$.

Q: A function $f$ has a local minimum at $x = 1$ if:

$f(1) \leq f(x)$ for all $x$ near $1$.

Q: Suppose the graph of $f$ is symmetric with respect to the origin. Which of the following must be true?

$f(-x) = -f(x)$ for all $x$ in the domain.

Q: A real-world scenario: The temperature $T$ (in $^\\circ$C) in a city over a day is modeled by a function $T(t)$, where $t$ is the hour. If $T(t)$ has an absolute maximum at $t = 15$ and an absolute minimum at $t = 5$, what does this mean?

The highest temperature occurs at 3 PM, and the lowest at 5 AM.

Q: Given the function $f(x) = x^2 + 6x + 1$, determine if it is even, odd, or neither.

Neither, because $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$.

Q: If a function $f$ is increasing on the interval $(-2, 4)$, which of the following is true?

For any $x_1 < x_2$ in $(-2, 4)$, $f(x_1) < f(x_2)$.

Q: A function $f$ is defined on $[-4, 6]$ and its graph has local maxima at $x = -2$ and $x = 4$. Which of the following best describes a local maximum?

A point where $f(x)$ is greater than or equal to all nearby values of $f$.

Question 1

Which of the following statements is TRUE about an even function $f(x)$?

Accepted Answer

It is symmetric with respect to the y-axis, so $f(-x) = f(x)$ for all $x$ in the domain.

Question 2

Given the function $f(x) = 2x^2 + 3$, determine whether it is even, odd, or neither.

Accepted Answer

Even, because $f(-x) = f(x)$ for all $x$.

Question 3

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

Accepted Answer

For any $x_1 < x_2$, $f(x_1) > f(x_2)$.

Question 4

Suppose a function $f$ is defined on $[-3, 5]$ and its graph has a highest point at $(2, 7)$ and a lowest point at $(-3, -2)$. What is the absolute maximum value of $f$?

Accepted Answer

$7$

Question 5

Which of the following functions is odd?

Accepted Answer

$f(x) = x^3 - x$

Question 6

A function $f$ is constant on an interval if:

Accepted Answer

For any $x_1 < x_2$, $f(x_1) = f(x_2)$.

Question 7

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

Accepted Answer

Odd, because $f(-x) = -f(x)$ for all $x$.

Question 8

A function $f$ has a local minimum at $x = 1$ if:

Accepted Answer

$f(1) \leq f(x)$ for all $x$ near $1$.

Question 9

Suppose the graph of $f$ is symmetric with respect to the origin. Which of the following must be true?

Accepted Answer

$f(-x) = -f(x)$ for all $x$ in the domain.

Question 10

A real-world scenario: The temperature $T$ (in $^\circ$C) in a city over a day is modeled by a function $T(t)$, where $t$ is the hour. If $T(t)$ has an absolute maximum at $t = 15$ and an absolute minimum at $t = 5$, what does this mean?

Accepted Answer

The highest temperature occurs at 3 PM, and the lowest at 5 AM.

Question 11

Given the function $f(x) = x^2 + 6x + 1$, determine if it is even, odd, or neither.

Accepted Answer

Neither, because $f(-x) 
eq f(x)$ and $f(-x) 
eq -f(x)$.

Question 12

If a function $f$ is increasing on the interval $(-2, 4)$, which of the following is true?

Accepted Answer

For any $x_1 < x_2$ in $(-2, 4)$, $f(x_1) < f(x_2)$.

Question 13

A function $f$ is defined on $[-4, 6]$ and its graph has local maxima at $x = -2$ and $x = 4$. Which of the following best describes a local maximum?

Accepted Answer

A point where $f(x)$ is greater than or equal to all nearby values of $f$.

Question 14

Suppose a function $f$ is constant on $[1, 3]$ and increasing on $[3, 5]$. Which of the following is true?

Accepted Answer

For $x_1, x_2$ in $[1, 3]$, $f(x_1) = f(x_2)$; for $x_1 < x_2$ in $[3, 5]$, $f(x_1) < f(x_2)$.

Question 15

A function $f$ is defined on $[-5, 5]$ and has an absolute minimum at $x = -5$ and an absolute maximum at $x = 2$. Which of the following is correct?

Accepted Answer

$f(-5)$ is the smallest value and $f(2)$ is the largest value of $f$ on $[-5, 5]$.

Question 16

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

Accepted Answer

It is symmetric with respect to the y-axis.

Question 17

A function $f$ is defined on $[0, 10]$ and is decreasing on $[0, 4]$, constant on $[4, 7]$, and increasing on $[7, 10]$. Which interval contains the absolute minimum?

Accepted Answer

$[0, 4]$

Question 18

If $f(x)$ is an even function and $f(3) = 5$, what is $f(-3)$?

Accepted Answer

$5$

Question 19

A function $f$ is defined on $[-2, 2]$ and has a local maximum at $x = 0$. Which of the following must be true?

Accepted Answer

$f(0) \geq f(x)$ for all $x$ near $0$.

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

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