Let $\displaystyle f\left(x\right)=\frac{ax+b}{cx-d}\; \left(d\ne 0\right)$ and $\displaystyle g\left(x\right)=\frac{-2x+3}{x-1}$. If $f\left(g\left(x\right)\right)=x$, find $f\left(x\right)$.
$f\left(x\right)=$
For $\displaystyle f\left(x\right)=\frac{3x+a}{x+b}, f^{-1}\left(1\right)=3$ and $f^{-1}\left(-7\right)=-1$. Find the values of $a$ and $b$.
$a=$
$, b=$
Let $f\left(x\right)=2x+k$ and $\displaystyle g\left(x\right)=\frac{8x}{2x+3}$.
Find the value of $k$ such that $g\left(f\left(x\right)\right)$ is an self-inverse function.