Question

Let $f:R\to R$ be defined as $f\left(x\right)=10x+7$ find the function $g:R\to R$ such that $g\cdot f=f\cdot g=I_R$

Answer 1

$f\left(x\right)=10x+7$

$\text{Let}f\left(x\right)=y$

$\Rightarrow 10x+7=y$

$\Rightarrow x=\frac{y-7}{10}$

$g\left(x\right)=\frac{x-7}{10}$

Now$fog\left(x\right)=f\left[g\right(x\left)\right]$

$=f\left(\frac{x-7}{10}\right)$

$=10\times \left(\frac{x-7}{10}\right)+7$

$=x$

$=I\left(x\right)$

Alose go $f\left(x\right)=g(10x+7)$

$=g\frac{10x+7-7}{10}$

$=x$

$=f\left(x\right)$

Thus $f.g=g.f=IR$