Given the function f(x)=1 x+2 . Find the points of discontinuity … - Vidyadip

Question

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

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Answer

We have, $\text{f(x)}=\frac{1}{\text{x}+2}$
$\therefore\ \text{y}=\text{f}\big\{\text{f(x)}\big\}$
$=\text{f}\Big(\frac{1}{\text{x}+2}\Big)=\frac{1}{\frac{1}{\text{x}+2}+2}$
$=\frac{1}{1+2\text{x}+4}\cdot(\text{x}+2)=\frac{(\text{x}+2)}{(2\text{x}+5)}$
So, the function y will not be continuous at those points, where it is not defined as it is a rational function.
Therefore, $\text{y}=\frac{(\text{x}+2)}{(2\text{x}+5)}$ is not defined, when 2x + 5 = 0
$\therefore\ \text{x}=\frac{-5}{2}$
Hence, y is discontinuous at $\text{x}=\frac{-5}{2}$

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