Question

If $f:x\to 2x+p$ and $f^{-1}:\left(x\right)\to m(4x+3)$, where $p$ and $m$ are some constants, find the values of $p$ and $m$.

Answer 1

Given, $f\left(x\right)=2x+p$

$\Rightarrow y=2x+p\Rightarrow y-p=2x$

$\Rightarrow x=\frac{y-p}{2}$

$\Rightarrow f^{-1}\left(x\right)=\frac{x-p}{2}$ ....(i)

Given, $f^{-1}\left(x\right)=m(4x+3)=4mx+3m$ ....(ii)

Comparing (i) and (ii), we get

$\frac{x}{2}-\frac{p}{2}=4mx+3m\Rightarrow 4m=\frac{1}{2}\Rightarrow m=\frac{1}{8}\Rightarrow -\frac{p}{2}=3m\Rightarrow -\frac{p}{2}=\frac{3}{8}\Rightarrow p=-\frac{3}{4}$