Question

Given the function $f\left(x\right)=\frac{1}{x+2}.$ Find the points of discontinuity of the function $f\left(f\left(x\right)\right)$.

Answer 1

Clearly $f\left(x\right)=\frac{1}{x+2}$ is discontinuous at $x=-2$.

Also, it is not defined at $x=-2$.

Now,

$f\left(f\left(x\right)\right)=f\left(\frac{1}{x+2}\right)=\frac{1}{\frac{1}{x+2}+2}=\frac{x+2}{2x+5}$

Thus we observe that $f\left(f\left(x\right)\right)$ is discontinuous and undefined at $x=-\frac{5}{2}$

Hence $f\left(f\left(x\right)\right)$ is discontinuous at $x=-2$ and $x=-\frac{5}{2}$.