Q: Suppose $$g(x) = x^3$ + 3. Is$$ $$g$$ even, odd, or neither?

Neither, because $$g(-x) \neq g(x)$$ and $$g(-x) \neq -g(x)$$

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 $$f(x) = x^3$ + 4x$$, determine whether $$f is $$even, odd, or neither.

Accepted Answer

Odd, because $$f(-x) = -f(x)$$

Question 3

Which of the following best describes a function that is constant on an interval $$I$$?

Accepted Answer

For all $$x in$$ $$I$$, $$f(x) is $$the same value

Question 4

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

Accepted Answer

$$f(2) \geq f(x)$$ for all $$x in $$some open interval containing $$2$$

Question 5

A function $$f is $$defined on $$[0, 5]$$ and attains its largest value at $$x = 3$$ and its smallest value at $$x = 5. $$What are the absolute maximum and minimum values?

Accepted Answer

Absolute maximum: $$f(3)$$, Absolute minimum: $$f(5)$$

Question 6

Which of the following is true according to the Extreme Value Theorem?

Accepted Answer

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

Question 7

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

Accepted Answer

$$3$$

Question 8

If the average rate of change of $$f(x)$$ from $$x = a to$$ $$x = b is$$ $$m$$, what is the equation of the secant line through $$(a, f(a))$$ and $$(b, f(b))$$?

Accepted Answer

$$y = m(x - a) + f(a)$$

Question 9

Suppose $$g(x) = x^3$ + 3. Is$$ $$g$$ even, odd, or neither?

Accepted Answer

Neither, because $$g(-x) 
eq g(x)$$ and $$g(-x) 
eq -g(x)$$

Question 10

A function $$f is $$increasing on an interval $$I if$$:

Accepted Answer

For any $$x_1$$ \u003c $$x_2 in$$ $$I$$, $$f(x_1$) \u003c f(x_2$)$$

Question 11

Given $$F(x) = 1 + |x|$$, determine if $$F is $$even, odd, or neither.

Accepted Answer

Even, because $$F(-x) = F(x)$$

Question 12

A company tracks the temperature of a chemical reaction over time. If the temperature function $$T(t)$$ has a local minimum at $$t = 5$$ minutes, what does this mean?

Accepted Answer

The temperature is lowest at $$t = 5$$ minutes compared to times near $$t = 5$$ minutes.

Question 13

If $$f(x) is $$decreasing on $$[0, 3]$$, which of the following must be true?

Accepted Answer

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

Question 14

A function $$f is $$defined on $$[1, 4]$$ except at $$x = 2$$, where there is a hole. If $$f(1) = 3$$, $$f(4) = 4$$, and $$f(2) is $$undefined, what can you conclude?

Accepted Answer

There is no absolute maximum, but the absolute minimum is $$f(1) = 3.$$

Question 15

Given $$f(x) = -2x^3$ + 5x^2$ + 1$$, use a graphing utility to estimate the $$x-$$value where the local maximum occurs on $$[-3, 3].$$

Accepted Answer

$$x \approx 1.67$$

Question 16

The average rate of change of $$f(x)$$ from $$x = a to$$ $$x = b is $$given by:

Accepted Answer

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

Question 17

A function $$f is $$defined on $$[2, 6]$$ and attains its minimum value at every $$x in$$ $$[2, 4]. $$What does this mean?

Accepted Answer

The function is constant on $$[2, 4]$$ and this value is the absolute minimum.

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

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