IB Mathematics Analysis and Approaches Question Type 6: Showing Whether a Function Is Self-inverse or Not Exercises

Show that the function $f(x)=\frac{1-x}{1+x}$ is self-inverse.

Show that $f(x)=\frac{3x-7}{7x-3}$ is a self-inverse function.

Show that $f(x)=\frac{2x+3}{3x-2}$ is its own inverse.

Show that $f(x)=\frac{2x-5}{5x-2}$ satisfies $f(f(x))=x$.

Verify that $f(x)=\frac{1}{x}$ is self-inverse for $x\ne 0$.

For $f(x)=\frac{ax+b}{cx+d}$ with $ad-bc\ne 0$ and $f(x)\not\equiv x$, show that $f$ is self-inverse if and only if $a+d=0$.

Show that the function $f(x)=\frac{x+5}{2x-1}$ is self-inverse.

Prove that the fractional linear function $f(x)=\frac{ax+b}{cx-a}$ is self-inverse provided $a^2+bc\ne 0$.

Find all real $p$ such that $f(x)=\frac{x+p}{px-1}$ is self-inverse.

Show that $f(x)=\frac{2-x}{1+2x}$ is its own inverse.

Show that the function $f(x)=5-x$ is self-inverse.

Determine all real $k$ such that $f(x)=\frac{kx+1}{x+k}$ is its own inverse.