Question

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

Answer 1

$f\left[f\left(x\right)\right]=\frac{1}{{\displaystyle \frac{1}{x+2}}+2}=\frac{x+2}{2x+5}$
$$\begin{gathered} \mathrm{So},f\left[f\left(x\right)\right]\mathrm{is}\mathrm{not}\mathrm{defined}\mathrm{at}x+2=0\mathrm{and}2x+5=0 \\ \mathrm{If}x+2=,\mathrm{then}x=-2 \\ \mathrm{If}2x+5=0,\mathrm{then}x=-\frac{5}{2} \end{gathered}$$
Hence, the function is discontinuous at $x=-\frac{5}{2}\mathrm{and}-2$