Inverse functions

Summary

• \(f: A→B\) is a given function
• If \(f\) is bijective then there exists a function denoted by

\[f^{-1}: B→A\]

• called the inverse of \(f\) .
• The domain of \(f^{-1}\) is the codomain of \(f\) and vice versa.
• For \(y∈B\) , \(f^{-1}(y)\) is equal to the unique value \(x∈A\) such that \(f(x)=y\) .
• Results:
• \(f^{-1}\left( f\left( x \right) \right)=x\) for all \(x∈A\) .
• \(f\left( f^{-1}\left( y \right) \right)=y\) for all \(y∈B\) .
• \(f^{-1}\) is bijective and its inverse is \(f\) .
• Examples:
• \(f:R→R\) defined by \(f\left( x \right)=ax+b\) for \(a≠0\) has the inverse

\[f^{-1}\left( y \right)=x= \frac{y-b}{a}\]

• where \(f^{-1}:R→R\) .
• \(f:[0,∞)→[0,∞)\) defined by \(f(x)=x^2\) has the inverse \(f^{-1}\left( y \right)=x=\sqrt{y}\) where \(f^{-1}:[0,∞)→[0,∞)\) .
• \(f:R→(0,∞)\) defined by \(f(x)=e^x\) has the inverse \(f^{-1}\left( y \right)=x=log y\) where \(f^{-1}:(0,∞)→R\) .