Question

Which of the following is true for a real valued function $y=f\left(x\right)$, defined on $[-a,a]$?

• A. $f\left(x\right)$ can be expressed as a sum or a difference of two even functions
• B. $f\left(x\right)$ can be expressed as a sum or a difference of two odd functions
• C. $f\left(x\right)$ can be expressed as a sum or a difference of an odd and an even function
• D. $f\left(x\right)$ can never be expressed as a sum or a difference of an odd and an even function

Answer 1

The correct option is C $f\left(x\right)$ can be expressed as a sum or a difference of an odd and an even function

Let,

$y=f\left(x\right)$
$=\frac{1}{2}\left[f\right(x)+f(-x)+f(x)-f(-x\left)\right]$
$=\frac{f\left(x\right)+f(-x)}{2}+\frac{f\left(x\right)-f(-x)}{2}$
Here,

$\frac{f\left(x\right)+f(-x)}{2}=$ even function
$\frac{f\left(x\right)-f(-x)}{2}=$odd function.
Hence $f\left(x\right)$ can be expressed as a sum or a difference of an odd and an even function.