This section discusses inverse functions and will cover the following:
Before we can learn to do any of this, however, let’s refresh your memory on the definition of a function!
A function is a relation in which each \(x\) value (input) has only one \(y\) value (output). Functions can be represented as ordered pairs, as equations, and as graphs.
The vertical line test can be used to determine if a graph represents a function.
Vertical Line Test: If vertical lines drawn through the graph intersect the graph in exactly one point, the graph represents a function. (Vertical lines are the \(x\) values, so if the \(x\) value is paired with only one \(y\) value, this satisfies the definition that each \(x\) value has only one \(y\) value).
- Graphing a function and its inverse.
- Determining if a function’s inverse is also a function.
- Writing the inverse of a function.
- Verifying (using composites) that two functions are inverses of each other.
Before we can learn to do any of this, however, let’s refresh your memory on the definition of a function!
A function is a relation in which each \(x\) value (input) has only one \(y\) value (output). Functions can be represented as ordered pairs, as equations, and as graphs.
The vertical line test can be used to determine if a graph represents a function.
Vertical Line Test: If vertical lines drawn through the graph intersect the graph in exactly one point, the graph represents a function. (Vertical lines are the \(x\) values, so if the \(x\) value is paired with only one \(y\) value, this satisfies the definition that each \(x\) value has only one \(y\) value).
Now that we have reviewed functions, let’s talk about inverse functions.
Let’s look at the inverse of the cubic function of the form \(f\left(x\right)={x}^{3}\). Move either the slider \(p\) or the point on the graph. What do you notice? What do you wonder?
The point on the graph has been reflected in the line \(y=x\). Note that the segment perpendicular to the line \(y=x\) (connecting the points on the function \(f\left(x\right)=x^3\) and its inverse) is bisected by the line \(y=x\). To see the inverse function, click on the circle next to the equation \(g\left(y\right)=f\left(y\right)\).
Let’s look at the inverse of the cubic function of the form \(f\left(x\right)={x}^{3}\). Move either the slider \(p\) or the point on the graph. What do you notice? What do you wonder?
The point on the graph has been reflected in the line \(y=x\). Note that the segment perpendicular to the line \(y=x\) (connecting the points on the function \(f\left(x\right)=x^3\) and its inverse) is bisected by the line \(y=x\). To see the inverse function, click on the circle next to the equation \(g\left(y\right)=f\left(y\right)\).
Quick Check 2
1) How are the two graphs related to the line \(y=x\)?
2) What do you notice about the points on the function and the points on the inverse?
3) What do you notice about the domain of the function and the range of the inverse?
4) What do you notice about the range of the function and the domain of the inverse?
Quick Check Solutions
1) How are the two graphs related to the line \(y=x\)?
2) What do you notice about the points on the function and the points on the inverse?
3) What do you notice about the domain of the function and the range of the inverse?
4) What do you notice about the range of the function and the domain of the inverse?
Quick Check Solutions
When looking at the graph of a function, you can determine if its inverse is also a function by using the horizontal line test.
Horizontal Line Test: If a horizontal line drawn through the graph of a function intersects the graph in exactly one point, then the inverse of the function is also a function.
Horizontal Line Test: If a horizontal line drawn through the graph of a function intersects the graph in exactly one point, then the inverse of the function is also a function.
Graphing Inverse Functions
To graph inverse functions:
Example 1: Graph the function \(f\left(x\right)=x^2\) and its inverse on the same coordinate graph. Determine if the inverse is a function.
To graph inverse functions:
- Make a table of values for the function or find the \(\left(x,y\right)\) coordinates from a given graph.
- Switch the \(\left(x,y\right)\) coordinates and plot them.
Example 1: Graph the function \(f\left(x\right)=x^2\) and its inverse on the same coordinate graph. Determine if the inverse is a function.
Writing Inverse Functions
To write the inverse of a function, follow these steps:
- Replace \(f\left(x\right)\) with \(y\).
- Interchange the \(x\) and \(y\) variables.
- Solve for \(y\).
- Rewrite using function notation--the notation for inverse functions is \(f^{-1}\left(x\right)\).
Example 2: Write the inverse of the function \(f\left(x\right)=\frac{2}{3}x-7\)
\(\begin{align}f\left(x\right)&=\frac{2}{3}x-7\ & \ &\text{1) Rewrite the original problem.}\\
y&=\frac{2}{3}x-7\ & \ &\text{2) Replace} f\left(x\right)\ \text{with y.}\\
x&=\frac{2}{3}y-7\ & \ &\text{3) Interchange the x and y variables.}\\
x+7&=\frac{2}{3}y\ & \ &\text{4) Add 7 to both sides to isolate the variable term.}\\
\frac{3}{2}\left(x+7\right)&=y\ & \ &\text{5) Multiply both sides by the reciprocal to isolate y.}\\
\frac{3}{2}x+\frac{21}{2}&=y\ & \ &\text{6) Distribute the fraction to each term.}\\
f^{-1}\left(x\right)&=\frac{3}{2}x+\frac{21}{2}\ & \ &\text{7) Replace y with inverse function notation.}\end{align}\)
y&=\frac{2}{3}x-7\ & \ &\text{2) Replace} f\left(x\right)\ \text{with y.}\\
x&=\frac{2}{3}y-7\ & \ &\text{3) Interchange the x and y variables.}\\
x+7&=\frac{2}{3}y\ & \ &\text{4) Add 7 to both sides to isolate the variable term.}\\
\frac{3}{2}\left(x+7\right)&=y\ & \ &\text{5) Multiply both sides by the reciprocal to isolate y.}\\
\frac{3}{2}x+\frac{21}{2}&=y\ & \ &\text{6) Distribute the fraction to each term.}\\
f^{-1}\left(x\right)&=\frac{3}{2}x+\frac{21}{2}\ & \ &\text{7) Replace y with inverse function notation.}\end{align}\)
Example 3: Write the inverse of the function \(f\left(x\right)=3x^2-1;\ x\ge0\)
Example 4: Write the inverse of the function \(f\left(x\right)=\sqrt[3]{x-4}+1\)
Verify Two Functions are Inverses of Each Other
Composition of functions can be used to verify if two functions are inverses of each other. Often times the directions will state “Use the composition of functions to determine if the functions are inverses of each other” or “Determine algebraically….” or “Use the definition of inverses….”
Given two functions \(f\left(x\right)\) and \(g\left(x\right)\), if \(f\left(g\left(x\right)\right)=x\) and \(g\left(f\left(x\right)\right)=x\) then the two function are inverses of each other.
It is important to recognize that the two composites must simplify to \(x\). It is not that they equal the same expression--they must both equal \(x\).
Example 5: Use composition of functions to verify that \(f\left(x\right)=5x+1\) and \(g\left(x\right)=\frac{\left(x-1\right)}{5}\) are inverses.
Example 5: Use composition of functions to verify that \(f\left(x\right)=5x+1\) and \(g\left(x\right)=\frac{\left(x-1\right)}{5}\) are inverses.
Example 6: Use composition of functions to verify that \(f\left(x\right)=4x^3-3\) and \(g\left(x\right)=\sqrt[3]{\frac{\left(x+3\right)}{4}}\) are inverses.