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 $$x_1$$ \u003c $$x_2 in$$ $$[1, 3]$$?

Accepted Answer

$$f(x_1$) \u003e f(x_2$)$$

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 $$x_1$$ \u003c $$x_2 in$$ $$[a, b]$$?

Accepted Answer

$$f(x_1$) \u003c f(x_2$)$$

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

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