Question

For the function $f:R\to R,f\left(x\right)=3x+2$, if $f^{-1}$, exists then find $f^{-1}\left(x\right)$.

Answer 1

Inverse of a function exists if it is both one-one and onto.

Checking for one-one:

if $f\left(x_1\right)=f\left(x_2\right)$

$⟹3x_1+2=3x_2+2$

$⟹x_1=x_2$

Hence , $f\left(x\right)$ is one-one

Co-domain of $f\left(x\right)$ is $R$

And, we know that $3x+2$ is a straight line graph and hence its range is $R$

Since, range $=$ co-domain , hence $f\left(x\right)$ is onto as well.

Hence, $f\left(x\right)$ is invertible.

Now, $y=f\left(x\right)=3x+2$

$⟹x=\frac{y-2}{3}$

And, $x=f^{-1}\left(y\right)$

$⟹f^{-1}\left(y\right)=\frac{y-2}{3}$

Replacing $y$ with $x$-

$⟹f^{-1}\left(x\right)=\frac{x-2}{3}$