Question

If the graph of the function of $y=f\left(x\right)$ is symmetrical about the line $x=2$, then

• A. $f\left(x+2\right)=f\left(x-2\right)$
• B. $f\left(2+x\right)=f\left(2-x\right)$
• C. $f\left(x\right)=f\left(-x\right)$
• D. $f\left(x\right)=-f\left(-x\right)$

Answer 1

The correct option is B

$f\left(2+x\right)=f\left(2-x\right)$

Explanation for the correct option:

As per the definition of even function, the graph of an even function is symmetric about the $y-$axis, so, $x=0$.

Thus, $g\left(x\right)=g\left(-x\right)$

$g\left(0+x\right)=g\left(0-x\right)$.

Therefore, if the graph of a given function is symmetric about $x=a$, then $g\left(a+x\right)=g\left(a-x\right)$.

The graph of given function, $y=f\left(x\right)$ is symmetrical about the line $x=2$.

Therefore, $f\left(2+x\right)=f\left(2-x\right)$.

Hence, option (B) $f\left(2+x\right)=f\left(2-x\right)$ is the correct answer.