Question

If $f$ is an even function defined on the interval $(-5,5)$ then find the total number of real value of $x$ satisfying the equations $f\left(x\right)=f\left(\frac{x+1}{x+2}\right)$ are

• A. $2$
• B. $4$
• C. $1$
• D. $8$

Answer 1

Answer: (A)

If $f\left(x\right)$ is an even function in $x\in (-5,5)$

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

then $f\left(x\right)=f\left(\frac{x+1}{x+2}\right)$ can be written

as $f\left(x\right)=f\left(\frac{1-x}{2-x}\right)$

By putting $x=-x$

$x=\frac{1-x}{2-x}$

$2x-x^2=1-x$

$x^2+1-3x=0$

$x=\frac{3\pm \sqrt{5}}{2}$

Therefore, two real values of $x$.

Hence, this is the answer.