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

$$f(x_1$) = f(x_2$)$$ \text{ for any } $$x_1$$, $$x_2$$ \in I$

Q: A function $$f is $$said to have a relative maximum at $$x = a if$$:

$$f(a) \u003e f(x)$$ \text{ for all } x \neq a \text{ in some open interval around } a$

Q: If a function $$f is $$constant on the interval $$(1, 3)$$, which of the following statements is true?

$$f(x_1$) = f(x_2$)$$ \text{ for any } $$x_1$$, $$x_2$$ \in (1, 3)$

Q: If a function $$f$$ has a relative minimum at $$x = b$$ and a relative maximum at $$x = a$$, which of the following is true?

$$f(b) \u003c f(x)$$ \text{ for all } x \neq b \text{ near } b$

Question 1

Given the function $$f(x) = (x+3)(x-1)^2$(x-2) as $$shown in the first image, on which interval is the function increasing?

Accepted Answer

$$(2, \infty)$$

Question 2

For the function in the second image, what is the interval where the function is decreasing?

Accepted Answer

$$(-5, -2)$$

Question 3

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

Accepted Answer

$$f(x_1$) = f(x_2$)$$ 	ext{ for any } $$x_1$$, $$x_2$$ \in I$

Question 4

A function $$f is $$said to have a relative maximum at $$x = a if$$:

Accepted Answer

$$f(a) \u003e f(x)$$ 	ext{ for all } x 
eq a 	ext{ in some open interval around } a$

Question 5

Which algebraic test would you use to determine if the graph of $$y = x^2$ - 1 is $$symmetric with respect to the $$y-$$axis?

Accepted Answer

Replace $$x$$ with $$-x$$ and check if the equation is unchanged

Question 6

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

Accepted Answer

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

Question 7

Given the function $$h(x) = x^5$ + 1$$, determine if it is even, odd, or neither.

Accepted Answer

Neither

Question 8

For the function $$q(x) = x^3$ + 3x$$, what type of symmetry does its graph have?

Accepted Answer

Origin symmetry (odd function)

Question 9

Suppose a function $$f is $$decreasing on the interval $$(-2, 0)$$, constant on $$(0, 2)$$, and increasing on $$(2, 4). $$Which of the following is true?

Accepted Answer

$$f$$ has a relative minimum at $$x = 2$$

Question 10

If a function $$f$$ has $$f(x) \u003c 0$$ for $$x in $$the interval $$(-3, 1)$$, what does this mean about the graph of $$f$$?

Accepted Answer

The graph is below the $$x-$$axis for $$x in$$ $$(-3, 1)$$

Question 11

Which of the following is the correct algebraic test for origin symmetry?

Accepted Answer

Replace $$x$$ with $$-x$$ and $$y$$ with $$-y$$ and check if the equation is unchanged

Question 12

A real-world scenario: The temperature $$T (in $$degrees Celsius) at time $$t (in $$hours) is modeled by a function $$T(t). If$$ $$T is $$increasing on $$(6, 12)$$, what does this mean?

Accepted Answer

The temperature rises between 6 AM and 12 PM

Question 13

Given the graph of $$f(x) in $$the first image, what are the $$x-$$intercepts?

Accepted Answer

$$x = -3, 1, 2$$

Question 14

If a function $$f is $$constant on the interval $$(1, 3)$$, which of the following statements is true?

Accepted Answer

$$f(x_1$) = f(x_2$)$$ 	ext{ for any } $$x_1$$, $$x_2$$ \in (1, 3)$

Question 15

Which of the following is NOT a correct way to describe where a function is increasing, decreasing, or constant?

Accepted Answer

Using closed intervals

Question 16

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

Accepted Answer

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

Question 17

Suppose $$f(x) is $$decreasing on $$(-\infty, 0)$$ and increasing on $$(0, \infty). $$What can you conclude about $$x = 0$$?

Accepted Answer

$$x = 0 is a $$relative minimum

Question 18

Given the function $$f(x) = x^2$ - 1$$, which of the following is true?

Accepted Answer

$$f(x) is $$even

Question 19

If a function $$f$$ has a relative minimum at $$x = b$$ and a relative maximum at $$x = a$$, which of the following is true?

Accepted Answer

$$f(b) \u003c f(x)$$ 	ext{ for all } x 
eq b 	ext{ near } b$

More on Functions and Their Graphs: Increasing/Decreasing Intervals, Extrema, and Symmetry

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