Inverse of a Function

Pronunciation: /ˈɪn.vɜrs ʌv ɑ fʌŋk.ʃən/ Explain

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.

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

Stated mathematically:

Understanding 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) = -1	g(-1) = 12	Yes No Correct. Since the input of each function is the output of the other function, the functions are inverses of each other. Incorrect. Since the input of each function is the output of the other function, the functions are inverses of each other.
f(1.5) = 2	g(1.5) = 2	Yes No Incorrect. Since the input f is 1.5 and the output is 2, and the input of g is 1.5, these two functions are not inverses of each other.Correct. Since the input f is 1.5 and the output is 2, and the input of g is 1.5, these two functions are not inverses of each other.

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. Notice that as you move the point along the line in Manipulative 4, 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.

How To Find the Inverse of a Linear Equation

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

References

1. McAdams, David E.. All Math Words Dictionary, inverse of a function. 2nd Classroom edition 20150108-4799968. pg 101-102. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Inverse of a Function. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 8/7/2018. http://www.merriam-webster.com/dictionary/inverse function. Buy the book
3. 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 8/7/2018. 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. 4/23/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/i/inverseofafunction.html.

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Revision History

4/23/2019: Updated equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
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.)