# Let f : $R \to R$ be the function defined by $f(x)=2x-3$ $\forall x \in R$.Write $f^{-1}.$

$\begin{array}{1 1} \large\frac{x-3}{2} \\\large\frac{2x}{3} \\ \large\frac{3x}{2} \\ \large\frac{x+3}{2} \end{array}$

Toolbox:
• $f:R \to R;$ A function $g:R \to R$ is inverse of f if $fog=I_R=gof$ when I is the identify function
$f:R \to R ; f(x)=2x-3$

we define a function

$y:R \to R$

Let y be an arbitary element of range f

$y=2x-3$

$=> x=\frac{y+3}{2}$

Let $g(y)=\frac{y+3}{2}$

$(gof) (x)=g(f(x))=g(2x-3)$

$=\frac{2x-3+3}{2}$

=x

Also $(fog)(y)=f(\frac{y+3}{2})$

$=2(\frac{y+3}{2})-3$

=y

Hence g is in inverse of f

and $f^{-1}(x)=g(x)=\frac{x+3}{2}$