# If $$f(x) = \frac { (4x+3) } { (6x-4) }, x \neq \frac {2} {3}$$, show that $f(x) =x$ ,for all $$x \neq \frac {2} {3}$$. What is the inverse of $$f$$

$\begin{array}{1 1} f \; itself \\ x^{-1} \\\big(\large \frac { (4x+3) } { (6x+4) }\big)^{-1}\\ \large \frac {(6x-4) }{(4x+3)}\end{array}$

Toolbox:
• A function $g$ is called inverse of $f$, if there exists $g:y \to x$ such that $g of =I_x$ and $fog=I_y$, where $I_x$ identify function in $x$ and $I_y$ identify function in $y$
• Given two functions $f:A \to B$ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \to C$ by $gof (x)=g(f(x))\;for\; all \;x \in A$
Given function $f$ defined by $f(x)=\large \frac{4x+3}{6x-4}$,$\;x \neq \frac{2}{3}$
Step 1: Calculating $fof$:
We know that given two functions $f:A \to B$ and $g:B \to C$, then composition of $f$ and $g$, $gof:A \to C$ by $gof (x)=g(f(x))\;for\; all \;x \in A$
$\Rightarrow (fof) (x)=f(f(x)) =f\large (\frac{4x+3}{6x-4})$
$\Rightarrow (fof) (x) =\Large\frac{4(\frac{4x+3}{6x-4})+3}{6(\frac{4x+3}{6x-4})-4}\;$=$\;\large \frac{16x+12+18x-12}{24x+18-24x+16}=\frac{34x}{34}=x$
Therefore $(fof)(x)=x$
Step 2: Calculating the inverse of $f$:
We see from Step1 above that $(fof)(x)=x$ and therefore $fof=I$
By definition of inverse of functions we conclude that the given function is invertible and the inverse of $f$ is itself.
edited Mar 19, 2013