Question

If $f$ is odd function , $g$ be an even function and $g\left(x\right)=f(x+5)$ then $f(x-5)$ equals

• A. $-g\left(x\right)$
• B. $g\left(x\right)$
• C. $f\left(5\right)$
• D. None of these

Answer 1

The correct option is A $-g\left(x\right)$

Since $f\left(x\right)$ is an odd function, $f\left(x-5\right)=-f\left(-x+5\right)$
Also $g\left(x\right)=f\left(x+5\right)$,
$\therefore g\left(-x\right)=f\left(-x+5\right)=-f\left(x-5\right)$
Hence, $f\left(x-5\right)=-g\left(-x\right)=-g\left(x\right)$, as $g\left(x\right)$ is even.