Question

If $f:x\to p+qx$,$f^{-1}\left(9\right)=(-1)$ and $f\left(3\right)=(-11)$. find the value of $p$ and $q$.

Answer 1

Given, $f\left(x\right)=p+qx$, $f^{-1}\left(9\right)=-1,f\left(3\right)=-11$

$\Rightarrow y=p+qx\Rightarrow qx=y-p\Rightarrow x=\frac{y-p}{q}$

Interchanging $x$ and $y$, we get

$f^{-1}\left(x\right)=\frac{x-p}{q}$

$\Rightarrow f^{-1}\left(9\right)=\frac{9-p}{q}\Rightarrow -1=\frac{9-p}{q}$

$\Rightarrow p-q=9$ .....(i)

$f\left(3\right)=-11$

$\Rightarrow p+3q=-11$ ....(ii)

subtract (i) from (ii)

$\Rightarrow p+3q-p+q=-11-9$

$\Rightarrow 4q=-20\Rightarrow q=-5$

sub in (ii)

$p=-11+3\left(5\right)=-11+15=4$

$\therefore p=4,q=-5$.