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Section2.9Composition and Inverse Functions

In the context of covarying variables, a function is a rule (operation) that predicts the value of the dependent variable (output) given the value of the independent variable (input). An inverse function is a function that predicts the reverse relation. What happens when these two functions are composed?

Suppose \(f : x \mapsto y\) has an inverse function \(f^{-1}: y \mapsto x\). The equivalent inverse equations are given by \begin{equation*}y=f(x) \quad \Leftrightarrow \quad x=f^{-1}(y).\end{equation*} When viewed as a map, these functions reverse one another's operations.

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When inverse operations are applied in sequence (composition), there is no net change. We call the function that does not change the input the identity function, \begin{equation*}\mathrm{id}(x) = x.\end{equation*} An alternate definition of an inverse function is through composition and the identity function.

Definition2.9.1

Suppose a function \(f\) defines the relationship \(x \mapsto y=f(x)\). A function \(g\) is the inverse function of \(f\) (and we write \(g=f^{-1}\) if \(f \circ g = \mathrm{id}\) and \(g \circ f= \mathrm{id}\). That is, \begin{gather*} g \circ f(x) = g(f(x)) = x \qquad (x \overset{f}{\mapsto} y \overset{g}{\mapsto} x),\\ f \circ g(y) = f(g(y)) =y \qquad (y \overset{g}{\mapsto} x \overset{f}{\mapsto} y). \end{gather*}

Recall that exponentials and logarithms are inverse functions and that \(\ln\) is the natural logarithm with base \(e\). We write \(\exp\) without a base to refer to the natural exponential, \(\exp(x) = e^x\). We often use the identity property of their compositions. \begin{gather*} \exp(\ln(x)) = e^{\ln(x)} = x, \\ \ln(\exp(x)) = \ln(e^x) = x. \end{gather*} It is important to recognize that it is not the symbols \(e\) and \(\ln\) that cancel. It is that the composition of inverse functions create the identity.

Example2.9.2

Find the inverse function for \(f(x) = 3x+4\). Then show that the compositions of \(f\) and \(f^{-1}\) are the identity function.

Solution
Example2.9.3

Find the inverse function for \(\displaystyle f(x) = \frac{2x}{x+1}\). Then show that the compositions of \(f\) and \(f^{-1}\) are the identity function.

Solution
Example2.9.4

Find the inverse function for \(f(x) = 2e^{x-5}\). Then show that the compositions of \(f\) and \(f^{-1}\) are the identity function.

Solution