Inverse of a Function

Pronunciation: /ɪnˈvɜrs/ Explain
There are two boxes. The top box has an arrow going in on the left. This arrow is labeled 'input'. The top box also has an arrow going out on the right. This arrow is labeled 'output'. The top box is labeled 'function - f(input) = output'. The bottom box has an arrow going in on the right. This arrow is labeled 'output becomes input'. The bottom box also has an arrow going out on the left. This arrow is labeled 'input becomes output'. The bottom box is labeled 'inverse - f(output) = input'.
Figure 1: A function and its inverse.

The inverse of a function is a relation which, given the output of the function, returns the input of the function.

FunctionInverse
f(x) = yf-1(y) = x
f(1.0) = 2.5f-1(2.5) = 1.0
f(2.2) = 3.7f-1(3.7) = 2.2
f(4.6) = 5.2f-1(5.2) = 4.6
f(6.8) = 9.7f-1(9.7) = 6.8
Table 2: Inverse of a function.

Stated mathematically:

f^(-1)(x) = y if and only if f(y)=x
Figure 1: Inverse of a function.

Check MarkUnderstanding Check

Given the function f and g, click the 'Yes' check box if they are inverses, or the 'No' check box if they are not.

f(12) = -1g(-1) = 12Yes No
f(1.5) = 2g(1.5) = 2Yes No

Graphs of Inverses of Functions

Click on the purple point and drag it to change the figure.

What is the geometric relationship between a function and its inverse?
Manipulative 1 - Inverse of a Function Created with GeoGebra.

When functions are inverses of each other, their graphs have a special relationship. Here is the graph of y=3x+2 and its inverse y=x/3-2/3. Notice that every point of each line is reflected across the line y=x to a corresponding point on the other line.

Click on the purple point in manipulative 4 and drag it to change the figure. Notice that as you move the point along the line, the coordinates of the point are inverses of the coordinates of the point on the inverse function.

Click on the blue point and drag it to change the figure.

Manipulative 2 - Draw the Inverse of a Function Created with GeoGebra.

Finding the Inverse of a Linear Equation

Steps to get the Inverse of a Linear Equation
StepResultJustification
1f(x)=2x-1Equation of which to find the inverse
2y=2x-1Change f(x) to y.
3y+1=2x-1+1Add 1 to both sides.
4y+1=2xSimplify
5(y+1)/2=2x/2Divide both sides by 2
6y/2+1/2=xSimplify
7y=x/2+1/2Swap the variables.
8f-1(x)=x/2+1/2Change back to function notation

References

  1. Inverse of a Function. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 8/6/2018. http://www.merriam-webster.com/dictionary/inverse function.
  2. Chrystal, G.. Introduction to Algebra for the use of Secondary School and Technical Colleges. 3rd edition. pp 68-70. www.archive.org. Adam and Charles Black. 1902. Last Accessed 8/6/2018. http://www.archive.org/stream/introductiontoal00chryuoft#page/68/mode/1up/search/inverse. Buy the book

Cite this article as:

McAdams, David E. Inverse of a Function. 8/7/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/i/inverseofafunction.html.

Image Credits

Revision History

8/6/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
3/2/2010: Added "References". (McAdams, David E.)
9/19/2008: Added figure 1, manipulative 1, and manipulative 2. (McAdams, David E.)
8/13/2008: Added 'More Information' and corrected step numbers in 'Finding an Inverse of a Linear Function'. (McAdams, David E.)
4/5/2008: Added color emphasis. Added understanding check (McAdams, David E.)
7/12/2007: Initial version. (McAdams, David E.)

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