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 $$f(-x) \neq -f(x).$$

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 $$x_1  f$$($$x_2$$)$$.

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 $$x_1$$ \u003c $$x_2$$, $$f(x_1$) = f(x_2$).$$

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 $$T (in$$ $$^\circC) in a $$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 $$x_1$$ \u003c $$x_2 in$$ $$(-2, 4)$$, $$f(x_1$) \u003c f(x_2$).$$

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 $$x_1$$ \u003c $$x_2 in$$ $$[3, 5]$$, $$f(x_1$) \u003c f(x_2$).$$

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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