Question

If a function $f:R\to R$ is defined by $f\left(x\right)=3x+4$ show that $f^{-1}$ (the inverse function of $f$) exists and find it.

Answer 1

Let, $y=f\left(x\right)=3x+4\Rightarrow f^{-1}\left(y\right)=x$

$\Rightarrow y=3x+4$

So, $y-4=3x$

$\frac{y-4}{3}=x=f^{-1}\left(y\right)$

$\therefore f^{-1}\left(x\right)=\frac{x-4}{3}$

Hence this is the inverse.

$f^{-1}$ exists $\forall x\in R$

Note: Function must be injective to be inversed.