Question:
If $f(x)=\frac{4 x+3}{6 x-4}, x \neq \frac{2}{3}$, show that $f \circ f(x)=x$ for all $x \neq \frac{2}{3}$. What is the inverse of $f$ ?
Solution:
$(f o f)(x)=f(f(x))$
$=f\left(\frac{4 x+3}{6 x-4}\right)$
$=\frac{4\left(\frac{4 x+3}{6 x-4}\right)+3}{6\left(\frac{4 x+3}{6 x-4}\right)-4}$
$=\frac{16 x+12+18 x-12}{24 x+18-24 x+16}$
$=\frac{34 x}{34}$
$=x$
$\Rightarrow(f o f)(x)=x=I_{X}$, where $I$ is an identity function.
So, $f=f^{-1}$
Hence, $f^{-1}=\frac{4 x+3}{6 x-4}$
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