Inverse Function Composition Property

For $f(x)=2*x+1$, if one incorrectly computes $f(f^{-1}(x))=\frac{x-1}{2}+1$, it fails to produce the identity function.

For $f(x)=2*x+1$ with inverse $f^{-1}(x)=\frac{x-1}{2}$, the composition $f(f^{-1}(x))=x$, which shows the inverse property.

For $f(x)=2*x+1$ with a mistakenly defined inverse $g(x)=2*x-1$, the composition does not yield the identity function.

For $f(x)=2*x+1$ and $g(x)=\frac{x}{2}+1$, the composite function does not return the original input.