If f(x) is an odd function, which of the following must be even?

Question

A. -f(x)

B. f(|x-1|)

C. none of these.

D. | f(x) | 

E. f(x+1)

Answer 1

Consider an odd function such as f(x) = x³. Remember that an odd function satisfies the following: f(-x) = -f(x) meaning that to the left of the origin, the graph goes down the same amount as it goes up to the right of the origin (shown in red in the graph below).

Now consider an even function such as x². Remember that an even function satisfies the following: f(-x) = f(x) meaning that the portion of the graph to the right of the origin is mirrored across the y-axis (as you move either right or left a certain amount, the y-values are the same). This is shown in blue on the graph below.

The way to get the red graph to look like the blue graph is to take the absolute value (answer D)