Question

Given the function $f\left(x\right)=1/(x+2)$. Find the points of discontinuity of the composite function $y=f\left(f\right(x\left)\right)$

Answer 1

Use the concept of continuity of composite functions We have $F\left(x\right)=\frac{1}{(x+2)}$
Clearly, $F\left(x\right)$ is discontinuous at $x=-2$ as the function$\frac{1}{(x+2)}$ is discontinuous when $x+2=0$
$\Rightarrow x=-2$ is not in domain of $F\left(F\right(x\left)\right)$,
hence$F\left(F\right(x\left)\right)$ is discontinuous at $x=-2$
Now, $F\left(x\right)=F\left(\frac{1}{(x+2)}\right)=\frac{1}{\frac{1}{x+2}+2}=\frac{x+2}{2x+5}$
$\Rightarrow F\left(F\right(x)$ is discontinuous at $x=\frac{-5}{2}$ as denominator should not be equal to $0$.
Hence, the points of discontinuity are $\frac{-5}{2}$ "and $–2$