Topics
4. Modeling with Functions
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United States of America
Grades 9-12

4.04 Inverses of linear functions

Lesson
Practice

Introduction

Inverses of linear functions are useful when modeling relationships. We explore and visualize inverse functions using reflection across a line of symmetry, a rigid transformation that we were introduced to in 8th grade. We can rewrite linear functions in different forms and use inverse operations to identify the inverse of a linear function algebraically.

Graphing inverses

Inverse functions can help us model relationships in the world around us. In this section, we will learn how to find the inverse of a linear function, and we will use a linear function to make sense of converting between units of temperature.

Inverse functions

Two functions f and g are said to be inverses when one function can be mapped to the other by a reflection across the line y=x. That is, for every point \left(x,y\right) in f, there is a point \left(y,x\right) in g.

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In the given figure, we have the graph of the line f\left(x\right) and it's reflection over the line y=x, labeled g\left(x\right).

x-6-4-20246
f\left(x\right)-3-2-10123

If we now create a table of values for f\left(x\right) andg\left(x\right), we can notice something about the relationship between the input and output pairs for each function.

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g\left(x\right)-6-4-20246

The x and y coordinates for g\left(x\right) are just the swapped around coordinate pairs of f\left(x\right).

So, we can see that if f\left(x\right) and g\left(x\right) are inverse linear functions, and f\left(x\right) has a domain of set X and a range of set Y, then g\left(x\right) will have a domain of set Y and a range of set X.

Exploration

Check the boxes to show points A and B, then use the m and b sliders to adjust the function.

Loading interactive...
  1. What happens when you check each of the other check boxes?
  2. What happens when you move the the m and b sliders again?

To graph the inverse of a function, take an ordered pair \left(x,y\right) from the function and swap the coordinates to \left(y,x\right). We only need two points to graph a line. The inverse of a linear function will always be a line reflected about the line y=x.

Examples

Example 1

Sketch the graph of the inverse of the given function, f\left(x\right)=\dfrac{3}{4}x:

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Worked Solution
Create a strategy

Graph the inverse by locating any two points on the line of the given function and inverting their coordinates.

Apply the idea

When x=4, f\left(4\right)=\dfrac{3}{4}\left(4\right)=3 so we know the line passes through \left(4,3\right).

When x=-4, f\left(-4\right)=\dfrac{3}{4}\left(-4\right)=-3 so we know the line also passes through \left(-4,-3\right).

To plot the inverse, we can reflect these points across the line y=x by switching the x and y coordinates. This gives us two points on the inverse: \left(3,4\right) and \left(-3,-4\right)

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Shown below is the graph of the inverse of f\left(x\right)=\dfrac{3}{4}x:

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Reflect and check

When sketching the inverse of a function on paper, you can simply check that the graph is the function's inverse by folding along the diagonal line y=x and confirming that the lines are a mirror reflection.

Example 2

Shown below is the graph of C=\dfrac{5}{9}\left(F-32\right), the function used to calculate the temperature in degrees Celsius when given the temperature in degrees Fahrenheit. The points \left(32,0\right) and \left(75, 24\right) are plotted.

Temperature conversion
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\text{Degrees Fahrenheit}
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\text{Degrees Celsius}
a

Graph the inverse of C=\dfrac{5}{9}\left(F-32\right).

Worked Solution
Create a strategy

In order to plot the inverse of C=\dfrac{5}{9}\left(F-32\right), we can use the ordered pairs plotted on the graph and switch them to \left(C,F\right).

Apply the idea

Plot the points \left(0, 32\right) and \left(24,75\right), then connect them with a straight line through the graph.

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\text{Degrees Celsius}
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\text{Degrees Fahrenheit}
Reflect and check

Notice that the axis labels changed when we plotted the inverse function.

b

Interpret the meaning of the inverse function, F.

Worked Solution
Create a strategy

Recall that graphing the inverse of a function involves inverting the coordinates of the ordered pairs of the function.

Apply the idea

Since the ordered pairs of the inverse function are \left(C, F\right), the inverse function must be used to calculate the temperature in degrees Fahrenheit when given a temperature in degrees Celsius.

Idea summary

We can graph the inverse of a function by reflecting the function about the line y=x. We do this by locating two points on the line of a given function and inverting their coordinates, then plotting the two new points and connecting them with a straight line.

Finding inverses algebraically

Every linear function has an inverse. We can find an inverse by graphing the line and reflecting it over the line y=x, but we can also solve for an inverse function algebraically.

The notation for the inverse of a function is as follows:

\displaystyle f\left(x\right) \to f^{-1}\left(x\right)
\bm{f\left(x\right)}
original function
\bm{f^{-1}\left(x\right)}
inverse function

To find an inverse algebraically, we need to:

  1. Identify the input (usually x) and the output (usually f\left(x\right) or y).
  2. Switch x and y.
  3. Solve for the new y.
  4. Write the inverse using f^{-1}\left(x\right) notation.

When we reflect a function over the line y=x, we are effectively switching the x and y-values. To find the inverse, we swap x and y, then solve for y to get f^{-1}\left(x\right).

Examples

Example 3

Consider the equtaion 3x-8y=48.

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Convert the equation to slope-intercept form and write it in function notation.

Worked Solution
Create a strategy

We will need to use properties of equality to manipulate the equation and solve for y.

Apply the idea
\displaystyle 3x-8y\displaystyle =\displaystyle 48Original equation
\displaystyle -8y\displaystyle =\displaystyle 48-3xSubtraction property of equality
\displaystyle y\displaystyle =\displaystyle -6+\dfrac{3}{8}xDivision property of equality
\displaystyle y\displaystyle =\displaystyle \dfrac{3}{8}x-6Commutative property
\displaystyle f\left(x\right)\displaystyle =\displaystyle \dfrac{3}{8}x-6Substitute y=f\left(x\right)
b

Find the inverse of the function from part (a).

Worked Solution
Create a strategy

Since the equation is in slope-intercept form and written with f\left(x\right), we can replace f\left(x\right) with y, swap x and y, and then solve for y.

Apply the idea
\displaystyle f(x)\displaystyle =\displaystyle \dfrac{3}{8}x-6Slope-intercept form from part (a)
\displaystyle y\displaystyle =\displaystyle \dfrac{3}{8}x-6Substitute f\left(x\right)=y
\displaystyle x\displaystyle =\displaystyle \dfrac{3}{8}y-6Switch x and y
\displaystyle x+6\displaystyle =\displaystyle \dfrac{3}{8}yAddition property of equality
\displaystyle 8\left(x+6\right)\displaystyle =\displaystyle 3yMultiplication property of equality
\displaystyle 8x+48\displaystyle =\displaystyle 3yDistributive property
\displaystyle \dfrac{8}{3}x+16\displaystyle =\displaystyle yDivision property of equality
\displaystyle y\displaystyle =\displaystyle \dfrac{8}{3}x+16Symmetric property of equality
\displaystyle f^{-1}(x)\displaystyle =\displaystyle \dfrac{8}{3}x+16Substitute y=f^{-1}\left(x\right)

The inverse of f\left(x\right)=\dfrac{3}{8}x-6 is f^{-1}\left(x\right)=\dfrac{8}{3}x+16.

Reflect and check

We can confirm that the two functions are inverses by graphing them on the same coordinate plane.

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Compare the slope of the linear function and its inverse.

Worked Solution
Create a strategy

We can find the slope of the linear function by considering the equation of the function in slope-intercept form.

Apply the idea

We can see that the slope of the original function f\left(x\right) is equal to \dfrac{3}{8} and the slope of the inverse f^{-1}\left(x\right) is equal to \dfrac{8}{3}. These values are reciprocals of one another.

Reflect and check

The slope of an inverse function is always equal to the reciprocal of the slope of the original linear function.

Example 4

Determine whether the function g\left(x\right)=\dfrac{x-5}{4}, represents the inverse of the function f\left(x\right), which is shown in the following table.

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f\left(x\right)5913172125
Worked Solution
Create a strategy

We can create a table of values by substituting each value of f\left(x\right) into g\left(x\right).

Apply the idea

Substituting each value of f\left(x\right) into g\left(x\right), we get:

x5913172125
g\left(x\right)012345

Since the output of g\left(x\right) when substituting in each value of f\left(x\right) is equal to the corresponding x-coordinate given in the original table of values, we can see that it is the inverse of f\left(x\right).

Reflect and check

We can test whether or not two functions are inverses by verifying that every point \left(x,y\right) in one function, is \left(y,x\right) in the other function.

Idea summary

The inverse of a function can be found algebraically by swapping the x and y variables and then solving for y.

Outcomes

F.BF.B.4

Find inverse functions.

F.BF.B.4.A

Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

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