Question

Given $f\left(x\right)=\frac{8}{x},x\ne 0$ and $\text{fog}\left(x\right)=4x$, find the (i) $g\left(x\right)$ (ii) the value of $x$, when $\text{gof}\left(x\right)=3$.

Answer 1

Given, $f\left(x\right)=\frac{8}{x}$ and $\text{fog}\left(x\right)=4x$

$f\left(g\right(x\left)\right)=\frac{8}{g\left(x\right)}\Rightarrow 4x=\frac{8}{g\left(x\right)}\Rightarrow g\left(x\right)=\frac{2}{x}$

Thus $g\left(f\right(x\left)\right)=\frac{2}{\frac{8}{x}}=\frac{x}{4}$

$\Rightarrow 3=\frac{x}{4}\Rightarrow x=12$