Question

Consider the function $f\left(x\right)=\{\begin{array}{c}\begin{array}{cc}\frac{x+5}{x-2}& \text{if}x\ne 2\\ 1& \text{if}x=2\end{array}\end{array}$. Then $f\left(f\right(x\left)\right)$ is discontinuous.

• A. At all real numbers
• B. At exactly two values of x
• C. At exactly one value of x
• D. At exactly three values of x

Answer 1

The correct option is B At exactly two values of x

$f\left(x\right)=\{\begin{array}{c}\frac{x+5}{x-2}ifx\ne 2\\ 1ifx=2\end{array}$

$f\left(f\left(x\right)\right)=\frac{\frac{x+5}{x-2}+5}{\frac{x+5}{x-2}-2}=\frac{6x-5}{-x+9}$

$f\left(f\left(x\right)\right)=\frac{6x-5}{9-x}$

At $x=9$ it is dicontinous and also given discontinous at $x=2$

So option $B$ is correct