Question Type 3: Finding the inverse function of simple functions Exercises

Find the inverse function of f(x)=rac{2x+5}{3x}\,.

Determine the inverse of $f(x)=\frac{3x-4}{5x+2}\,.$

Find $f^{-1}(x)$ for $f(x)=\frac{4x+1}{2x-3}\,.$

Find the inverse of $f(x)=\frac{x-2}{x+3}$ and state its domain and range.

For $f(x)=\frac{3x+7}{x-5}\,,$ (a) find $f^{-1}(x)$, (b) show that $f\bigl(f^{-1}(x)\bigr)=x\,.$

Given $f(x)=\frac{5x-3}{2x+1}\,,$ show that $f^{-1}\bigl(f(x)\bigr)=x\,.$

Given parameters $a,b,c$, define $f(x)=\frac{a}{x-b}+c\,.$
(a) Find $f^{-1}(x)$.
(b) Show that $f\bigl(f^{-1}(x)\bigr)=x$.

Let $f(x)=\frac{2x-1}{3x+4}\,.$ Prove that $f\bigl(f^{-1}(x)\bigr)=x$ and $f^{-1}\bigl(f(x)\bigr)=x\,.$

For $f(x)=\frac{2x+1}{x-k}\,,$ determine the value(s) of $k$ for which $f$ is its own inverse.

Let $f(x)=\frac{x+b}{bx-1}\,.$ Determine all real values of $b$ for which $f$ is its own inverse.

For the general linear fractional function $f(x)=\frac{px+q}{rx+s}\,,$ (a) derive a formula for the inverse $f^{-1}(x)$, (b) state the condition on $p,q,r,s$ for invertibility.