Question

If the unctions $f,g,h$ are defined from ${}^{\prime }{R}^{\prime }$ to ${}^{\prime }{R}^{\prime }$ by $f\left(x\right)=x^2-1,g\left(x\right)=\sqrt{x^2+1},h\left(x\right)=\{\begin{array}{c}0,\\ x,\end{array}\begin{array}{c}ifx\le 0\\ ifx\ge 0\end{array}$, then hofog is equal to:

• A. $x^2$
• B. $\sqrt{x^4-1}$
• C. $0$
• D. $noneofthese$

Answer 1

The correct option is A $x^2$

To find nofog , frist find fog

$f\left(x\right)=x^2-1;g\left(x\right)=\sqrt{x^2+1}$

$\Rightarrow f\left(g\right(x\left)\right)={\left(\sqrt{x^2+1}\right)}^2-1$

$\Rightarrow f\left(g\right(x\left)\right)=x^2+1-1=x^2$

Now to find hofog substitute the value so

obtained in $h\left(x\right)$

$\Rightarrow h\left(f\right(g\left(x\right)\left)\right)=x^2$

Hence, $\left(A\right)$ is the correct option.