Composition of Functions

Given two functions, $LaTeX: f\left(x\right)$ and $LaTeX: g\left(x\right)$ , the composition of and is denoted by $LaTeX: f\:\circ\:g$ and is defined as follows:

$LaTeX: \left(f\:\circ\:g\right)\left(x\right)\:=\:f\left(g\left(x\right)\right)$

For example, if $LaTeX: f\left(x\right)\:=\:x^2-9$ , and $LaTeX: g\left(x\right)\:=\:2x\:+\:5$ , then

$LaTeX: \left(f\:\circ\:g\right)\left(x\right)\:=\:f\left(2x\:+\:5\right)\:=\:\left(2x\:+\:5\right)^2\:-\:9\:$ $LaTeX: \:=\:4x^2\:+\:20x\:+\:25\:-\:9\:=\:4x^2\:+\:20x\:+\:16$

and $LaTeX: \left(g\:\circ\:f\right)\left(x\right)\:=g\:\left(x^2\:-\:9\right)\:$ $LaTeX: \:=\:2\left(x^2\:-\:9\right)\:+\:5\:=$ $LaTeX: 2x^2\:-\:18\:+\:5\:=\:2x^2\:-\:13$

Also, $LaTeX: \left(f\:\circ\:f\right)\left(x\right)\:=\:f\left(x^2\:-\:9\right)\:=\:\left(x^2\:-\:9\right)^2\:-\:9\:$ $LaTeX: =\:x^4\:-\:18x^2\:+\:81\:-\:9$ $LaTeX: =\:x^4\:-\:18x^2\:+\:72$

and $LaTeX: \left(g\:\circ\:g\right)\left(x\right)\:=g\:\left(2x\:+\:5\right)\:=\:$ $LaTeX: 2\left(2x\:+\:5\right)\:+\:5\:=\:4x\:+\:10\:+\:5\:=\:4x\:+\:15$

Inverse Functions

Given a function, $LaTeX: f\left(x\right)$ , an inverse function for is a function denoted by $LaTeX: f^{-1}\left(x\right)$ , such that

$LaTeX: \left(f^{-1}\:\circ\:f\right)\left(x\right)\:=\:x$ and $LaTeX: \left(f\:\circ\:f^{-1}\right)\left(x\right)\:=\:x$

In general terms, the function $LaTeX: f^{-1}$ undoes the action of and the function undoes the action of $LaTeX: f^{-1}$ .

For example, if $LaTeX: f\left(x\right)\:=\:2x\:-\:4$ , then $LaTeX: f^{-1}\left(x\right)\:=\:\frac{x+4}{2}$ .

Note that, $LaTeX: \left(f^{-1}\:\circ\:f\right)\left(x\right)\:=\:f^{-1}\left(2x\:-\:4\right)$ $LaTeX: \:=\:\frac{\left(2x\:-\:4\right)\:+\:4}{2}\:=\:\frac{2x}{2}\:=\:x$

and $LaTeX: \left(f\:\circ\:f^{-1}\right)\left(x\right)\:=\:f\left(\frac{x\:\:+\:4}{2}\right)\:$ $LaTeX: =\:2\left(\frac{x\:\:+\:4}{2}\right)\:-\:4\:=\:x\:+\:4\:-\:4\:=\:x$

Not all functions have an inverse function.

Definition: $LaTeX: f\left(x\right)$ is a one-to-one function if and only if whenever $LaTeX: f\left(a\right)\:=\:f\left(b\right)$ , then $LaTeX: a\:=\:b$ .

This means that for a one-to-one function, there cannot be two distinct input values that correspond to the same output value. Graphically, a function is a one-to-one function if and only if the graph of that function satisfies the horizontal line test.

Theorem: The function $LaTeX: f\left(x\right)$ has an inverse function, if and only if $LaTeX: f\left(x\right)$ is a one-to-one function.