Question

If the graph of $y=f\left(x\right)$ is symmetrical about the lines $x=1$ and $x=2$, then which of the following is true?

• A. $f(x+1)=f\left(x\right)$
• B. $f(x+3)=f\left(x\right)$
• C. $f(x+2)=f\left(x\right)$
• D. None of these

Answer 1

The correct option is C $f(x+2)=f\left(x\right)$

$f\left(x\right)$ is symmetrical about the line $x=1$

So,

$f(1-x)=f(1+x)$ .................... (1)

$f\left(x\right)$ is symmetrical about the line $x=2$

So,

$f(2-x)=f(2+x)$ .................... (2)

Replace $x$ with $1-x$ in equation (1) ,

$f(1-1+x)=f(1+1-x)$

$f\left(x\right)=f(2-x)$ ..........................(3)

From (2) and (3) ,

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