Definition 2.6.1.
A relation is a set of ordered pairs of the form \((x,y)\text{.}\) A function is a relation where each \(x\) value has exactly one output value \(y\text{.}\)
Section 2.6 Functions and Inverse Functions
Inverse functions are functions that undo the action of a given function.
Goals:
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F: Be able to determine the inverse of a function given given in any form (graph, table, equation).
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F: Be able to determine the domain and/or range of a function given as an equation or a graph.
Example 2.6.2.
Decide if each relation below, indicated by a graph or by a table, represents a function of \(x\)
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Graph:
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Graph:
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Table:
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Table:
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Table:
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Table:
Definition 2.6.3.
Given a function \(f\) defined on a domain \(D\text{,}\) a function \(g\) on a domain \(E\) is an inverse of \(f\) if \(f(g(x))=x\) whenever \(x\) is in the domain of \(g\text{,}\) and \(g(f(x))=x\) whenever \(x\) is in the domain of \(f\text{.}\) Often, the inverse function \(g\) is written \(f^{-1}\text{.}\)
Example 2.6.4.
On the domain of all real numbers, \(g(x)=\sqrt[3]{x}\) is the inverse of \(f(x)=x^3\text{.}\) Verify that \(g(x)\) and \(f(x)\) are inverses.
Investigation 2.6.1.
For the function \(r(x)\) as defined in Table 2.6.5, the inverse function \(r^{-1}(x)\) is the function given in Table 2.6.6.
| \(x\) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| \(r(x)\) | 25 | 60 | 1 | -10 |
| \(x\) | 25 | 60 | 1 | -10 |
|---|---|---|---|---|
| \(r^{-1}(x)\) | 0 | 1 | 2 | 3 |
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What do you notice about the two rows in the tables for the functions?
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Compute \(r^{-1}(r(1))\text{,}\) \(r(r^{-1}(1))\text{,}\) \(r^{-1}(r(2))\) and \(r(r^{-1}(25))\text{.}\)
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What are the domain and range of \(r\text{?}\)
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What are the domain and range of \(r^{-1}\text{?}\)
Investigation 2.6.2.
Consider the function \(f(x)=x^2\text{.}\)
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Complete the following table of output values of \(f\text{:}\)
Table 2.6.7. \(f\) \(x\) -2 -1 0 1 2 \(f(x)\) -
Create a table for a relation, \(g\text{,}\) that reverses the input and output of \(f\text{.}\) Is \(g\) a function?
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Since \(g\) is not a function, it cannot be the inverse of \(f\text{.}\) In order for a function to have an inverse, it must be one-to-one, that is every output has exactly one input. Sketch a function that is one-to-one and a function that is not one-to-one.
Figure 2.6.8. Sketches of one-to-one function and a function that is not one-to-one. -
Sometimes we can restrict the domain of a function to make it one-to-one, so that the restricted function will have an inverse. On what domain would \(f(x)=x^2\) have an inverse? What is the range of \(f\) on this domain?
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What is the inverse of \(f\) on the domain \([0, \infty)\text{?}\) What are the domain and range of \(f^{-1}\text{?}\)
Problem 2.6.9.
Refer to the function \(z(t)=\sqrt{t-3}\text{.}\)
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What are the domain and range of \(z\text{?}\)
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Find a formula for the inverse function, \(z^{-1}\text{.}\)
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What are the domain and range of \(z^{-1}\text{?}\)
Problem 2.6.10.
Refer to the function \(h(x)=3x-7\text{.}\)
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What are the domain and range of \(h\text{?}\)
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Find a formula for the inverse function, \(h^{-1}\text{.}\)
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What are the domain and range of \(h^{-1}\text{?}\)
Problem 2.6.11.
Refer to the functions \(z\) and \(z^{-1}\) in Problem 2.6.9, and \(h\) and \(h^{-1}\) in Problem 2.6.10.
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For each pair of functions, use Desmos to graph them on the same set of axes, along with the line \(y=x\text{.}\) Sketch each graph.
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What do you notice about the relationship between the graph of a function and the graph of its inverse? Explain why this happens.
Problem 2.6.12.
What do you know about the relationship between the domain and range of a function and the domain and range of its inverse?
