If f(x) is an odd function, which of the following must also be odd?A.|f(x)|B.- f(x)C.  f(|x|)D.None of theseE.  f(x - 1)SUBMITarrow_backPREVIOUS

To determine which of the given options must also be odd if $f(x)$ is an odd function, we need to recall the definition of an odd function. A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$.

Let's analyze each option:

A. $|f(x)|$

• The absolute value of $f(x)$ is not necessarily odd. For example, if $f(x) = x$, then $|f(x)| = |x|$, which is not an odd function because $|x| \neq -|x|$.

B. $-f(x)$

• If $f(x)$ is odd, then $-f(x)$ is also odd. This is because $-f(-x) = -(-f(x)) = f(x)$, which satisfies the condition for being an odd function.

C. $f(|x|)$

• The function $f(|x|)$ is not necessarily odd. For example, if $f(x) = x$, then $f(|x|) = |x|$, which is not an odd function because $f(-|x|) = |x| \neq -|x|$.

D. None of these

• This option suggests that none of the given functions must be odd if $f(x)$ is odd. However, we have already determined that $-f(x)$ must be odd.

E. $f(x - 1)$

• The function $f(x - 1)$ is not necessarily odd. For example, if $f(x) = x$, then $f(x - 1) = x - 1$, which is not an odd function because $f(-(x - 1)) = -(x - 1) \neq -(x - 1)$.

Therefore, the correct answer is:

B. $-f(x)$