Question

Given the function $f\left(x\right)=\frac{x-2}{5}$ and $\text{gof}\left(x\right)=x-3$. Find the component function $g$.

Answer 1

Given, $f\left(x\right)=\frac{x-2}{5}$

Thus $g\left(f\right(x\left)\right)=x-3$

$\Rightarrow g\left(\frac{x-2}{5}\right)=x-3$ ....(i)

$\Rightarrow \frac{x-2}{5}=z\Rightarrow x=5z+2$

Substituting in (i), we get

$\Rightarrow g\left(\frac{5z+2+2}{5}\right)=5z-1\Rightarrow g\left(z\right)=5(z-1)\Rightarrow g\left(x\right)=5x-1$