Inverse Functions

Definition of Inverse Function

LetLaTeX: ffbe a one-to-one function. Then LaTeX: gg is the invers function of LaTeX: ff if LaTeX: \left(f\circ g\right)\left(x\right)=x\:for\:every\:x\:in\:the\:domain\:of\:g,\:(f∘g)(x)=xforeveryxinthedomainofg,and LaTeX: \:\left(g\circ f\right)\left(x\right)=x\:for\:every\:x\:in\:the\:domain\:of\:f.(g∘f)(x)=xforeveryxinthedomainoff.

Example

Is LaTeX: gg the inverse ofLaTeX: ff if LaTeX: f\left(x\right)=x^3-1\:f(x)=x3−1and LaTeX: g\left(x\right)=\sqrt[3]{x+1}g(x)=3√x+1 ?

Solution

To determine whether or not LaTeX: g(x)g(x) is the inverse function ofLaTeX: f(x)f(x), we use the definition above:

LaTeX: \left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)=f\left(\sqrt[3]{x+1}\right)=\left(\sqrt[3]{x+1}\right)^3-1=x+1-1=x(f∘g)(x)=f(g(x))=f(3√x+1)=(3√x+1)3−1=x+1−1=x

and

LaTeX: \left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)=g\left(x^3-1\right)=\sqrt[3]{\left(x^3-1\right)+1}=\sqrt[3]{x^3}=x(g∘f)(x)=g(f(x))=g(x3−1)=3√(x3−1)+1=3√x3=x

Since, LaTeX: \left(f\circ g\right)\left(x\right)=\left(g\circ f\right)\left(x\right)=x,\:(f∘g)(x)=(g∘f)(x)=x,function LaTeX: gg is the inverse of function LaTeX: f.f.

Notation for Inverse Functions

The inverse function of the function LaTeX: f\left(x\right)\:f(x)is written as LaTeX: f^{-1}\left(x\right).f−1(x).

Example

If LaTeX: f\left(x\right)=x^3-1\:f(x)=x3−1, then LaTeX: f^{-1}\left(x\right)=\sqrt[3]{x+1}.f−1(x)=3√x+1.

Note: Do not confuse the -1 in LaTeX: f^{-1}\left(x\right)\:f−1(x)with a negative exponent. The symbol LaTeX: f^{-1}\left(x\right)\:f−1(x)represents the inverse function of LaTeX: f\left(x\right)\:f(x), NOT LaTeX: \frac{1}{f\left(x\right)}.1f(x).