Find the inverse of the following functions

i) if $a,b\in R,f:R\to R$ defined by $f\left(x\right)=ax+b(a\ne 0)$

ii) $f:\left(R\right)\to (0,\infty )$ defined by $f\left(x\right)=5^x$ 

iii) $f:(0,\infty )\to R$ defined by $f\left(x\right)={\log }_2^x$

iv) $f\left(x\right)=e^{4x+7}$

v) $f:R\to R,f\left(x\right)=\frac{2x+1}{3}$

i) Given $f\left(x\right)=a x+b$

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

$\Rightarrow ax+b=y \Rightarrow x=\frac{y-b}{a}$

$\Rightarrow f^{-1}\left(y\right)=\frac{y-b}{a}\therefore f^{-1}\left(x\right)=\frac{x-b}{a}$

ii) Given $f\left(x\right)=5^x$

let $f\left(x\right)=y\Rightarrow x=f^{-1}\left(y\right)$

$\Rightarrow 5^x=y\Rightarrow x={\log }_5^y$

$\Rightarrow f^{-1}\left(y\right)={\log }_5^y\therefore f^{-1}\left(x\right)={\log }_5^x$

iii) $f\left(x\right)={\log }_2^x$

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

$\Rightarrow {\log }_2^x=y\Rightarrow x=2^y$

$\Rightarrow f^{-1}\left(y\right)=2^y \therefore f^{-1}\left(x\right)=2^x$

iv) $f\left(x\right)=e^{4x+7}$

let $f\left(x\right)=y\Rightarrow x=f^{-1}\left(y\right)\Rightarrow e^{4x+7}=y$

Taking log on both sides

$\Rightarrow \log e^{4x+7}=\log y\Rightarrow 4x+7=\log y$

$\Rightarrow x=\frac{\log y-7}{4}\Rightarrow f^{-1}\left(y\right)=\frac{\log y-7}{4}$

$\therefore f^{-1}\left(x\right)=\frac{\log x-7}{4}$

v) $f\left(x\right)=\frac{2x+1}{3}$

Let $f\left(x\right)=y\Rightarrow x=f^{-1}\left(y\right)\Rightarrow \frac{2x+1}{3}=y$

$\Rightarrow f^{-1}\left(y\right)=\frac{3y-1}{2}\therefore f^{-1}\left(x\right)=\frac{3x-1}{2}$