Question

7. Which if the following is always true of odd functions?

I. f (−x) = −f(x)
II. f(|x|) is even
III. |f(x)| is even

A. All of these are true.
B. None of these are true.
C. I only.
D. II only.
E. I and III only.

Answer 1

A. All of these are true

Explanation:

I. Given the definiton of an odd function, f(-x) = -f(x), which is correct.

II. f(|x|) is always even, even if f(x) isn't odd. That is because f(|x|) is symmetric about the y-axis. Also, f(|-x|)=f(|x|), which is the definition of an even function. So, this is also correct.

III. |f(-x)| = |-f(x)| because remember, f(-x)=-f(x) because f(x) is odd. |-f(x)|=|f(x)|, so |f(-x)|=|f(x)|, which shows that is also an even function. This is correct.

Therefore, the answer is all of these are true.

Answer 2

f is an odd function if and only if I. f (−x) = −f(x)
C. I only.