Question

Show that function $f:R\to R,f\left(x\right)=\frac{2x+5}{8}$ is invertible. Also find inverse of f.

Answer 1

Given:-

$f\left(x\right)=\frac{2x+5}{8}f:R\to R$

Let $f\left(x\right)=f\left(y\right)$

$\frac{2x+5}{8}=\frac{2y+5}{8}$

$2x+5=2y+5$

$\Rightarrow x=y$

$\therefore f\left(x\right)$ is one-one.

Now,

Let $y=f\left(x\right)$

$y=\frac{2x+5}{8}$

$8y=2x+5$

$\Rightarrow x=\frac{8y-5}{2}$

$\Rightarrow g\left(x\right)=\frac{8y-5}{2}g:R\to R$

Hence $f\left(x\right)$ is one-one and onto.

$f^{-1}x=g\left(x\right)=\frac{8x-5}{2}$.