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:

Figure 1: Inverse of a function. 
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
 McAdams, David E.. All Math Words Dictionary, inverse of a function. 2nd Classroom edition 201501084799968. pg 101102. Life is a Story Problem LLC. January 8, 2015. Buy the book
 Inverse of a Function. merriamwebster.com. Encyclopedia Britannica. MerriamWebster. Last Accessed 8/7/2018. http://www.merriamwebster.com/dictionary/inverse function. Buy the book
 Chrystal, G.. Introduction to Algebra for the use of Secondary School and Technical Colleges. 3rd edition. pp 6870. 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
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.)