Question

Given, $f\left(x\right)=\left|x-1\right|,f:R\to R$ and $g\left(x\right)=e^x,g:\left(-1,\infty \right)\to R$. The function $fog\left(x\right)$ is defined, then its domain and range respectively are

• A. $\left(0,\infty \right)\&\left(0,\infty \right)$
• B. $\left(-1,\infty \right)\&\left(0,\infty \right)$
• C. $\left(-1,\infty \right)\&\left[1-\frac{1}{e},\infty \right]$
• D. $\left(-1,\infty \right)\&\left[\frac{1}{e},-1,\infty \right]$

Answer 1

The correct option is B $\left(-1,\infty \right)\&\left(0,\infty \right)$

Since, $f:R\to R$ defined by $f\left(x\right)=|x-1|$ and $g:(-1,\infty )\to R$ defined by $g\left(x\right)=e^x$

Thus, $fog\left(x\right)=f\left(g\right(x\left)\right)$

$\Rightarrow fog\left(x\right)=|e^x-1|$

$\therefore$ Domain $=(-1,\infty )$ and Range $=(0,\infty )$.