Inverse Functions

\(\blacksquare\) If \(f(x)=y\), then the inverse function is \(f^{–1}(y)=x\).

\(\blacksquare\) Only one-to-one functions have inverse functions.

\(\blacksquare\) \(f\) and \(g\) are inverse functions of each other if and only if \(fg(x)=x\), \(x\) in domain of \(g\) and\(gf(x)=x\), \(x\) in domain of \(f\).

\(\blacksquare\) If \(f\) and \(g\) are inverse functions of each other, then

(a) domain of \(f\)=range of \(g\) and domain of \(g\)=range of \(f\).

(b) graph \(g\) is the reflection of graph \(f\) on the line \(y=x\).

\(\blacksquare\) Horizontal line test can be used to test the existence of inverse functions.

\(\blacksquare\) \(ff^{–1}(x)=f^{–1}f(x)=x\)