Question

If $f\left(x\right)=\frac{1}{x-2}$ and $g\left(x\right)=\frac{3}{x-5}$, then the domain of $f\left(g\right(x\left)\right)$ is

• A. $R-\{2,5,\frac{13}{2}\}$
• B. $R-\{2,5\}$
• C. $R-\{5,\frac{13}{2}\}$
• D. $\varphi$

Answer 1

The correct option is C $R-\{5,\frac{13}{2}\}$

Given : $f\left(x\right)=\frac{1}{x-2},g\left(x\right)=\frac{3}{x-5}$
Domain of $f$ is $R-\left\{2\right\}$
Domain of $g$ is $R-\left\{5\right\}$

Clearly, when $g\left(x\right)$ is not defined, then $f\left(g\right(x\left)\right)$ is not defined.
So, $x=5$ does not lie in domain of $f\left(g\right(x\left)\right)$

Also, when $g\left(x\right)=2,$ $f\left(g\right(x\left)\right)$ is not defined.
$\frac{3}{x-5}=2$
$\Rightarrow x=\frac{13}{2}$

Therefore, the domain of $f\left(g\right(x\left)\right)$ is $R-\{5,\frac{13}{2}\}$