Question

If $f\left(x\right)=\frac{2x-1}{x-2}$
what is the inverse function of $f\left(x\right)$

Answer 1

Given,$f\left(x\right)=\frac{2x-1}{x-2}$

Substitute $y$ for$f\left(x\right)$.

$y=\frac{2x-1}{x-2}$

Swap the $x$ and $y$.

$x=\frac{2y-1}{y-2}$

Solve for $y$.

$x(y-2)=2y-1$

$xy-2x=2y-1$

$2y-xy=-2x+1$

Factor out y from the left side.

$y(2-x)=-2x+1$

$y=-2x+12-x$

Rewrite y as $f^{-1}\left(x\right)$.

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