Inverse of a Function
Pronunciation: /ɪnˈvɜrs/ ?

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 NoCorrect. 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 NoIncorrect. 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
 Manipulative 4: Inverse of a linear function. 

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/32/3.
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.

 Manipulative 5: Draw the inverse of a function. 

Click on the blue point in manipulative 5 and drag it to draw the inverse of the function
in the manipulative.
To change the function, right click on the blue line representing the
function. In the dropdown menu, click on 'Object Properties'. Then click on the
'Value' field. Type in a new function. Use '^' to show exponents.

Finding the Inverse of a Linear Equation
Steps to get the Inverse of a Linear Equation 

Step  Result  Justification 
1  f(x)=2x1  Equation of which to find the inverse 
2  y=2x1  Change f(x) to y. 
3  y+1=2x1+1  Add 1 to both sides. 
4  y+1=2x  Simplify 
5  (y+1)/2=2x/2  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
 Inverse of a Function. merriamwebster.com. Encyclopedia Britannica. (Accessed: 20100302). http://www.merriamwebster.com/dictionary/inverse function.
 Chrystal, G.. Introduction to Algebra for the use of Secondary School and Technical Colleges, 3rd edition, pp 6870. Adam and Charles Black, 1902. (Accessed: 20100302). http://www.archive.org/stream/introductiontoal00chryuoft#page/68/mode/1up/search/inverse.
Printed Resources
Cite this article as:
Inverse of a Function. 20100302. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/i/inverseofafunction.html.
Translations
Image Credits
Revision History
20100302: Added "References" (
McAdams, David.)
20080919: Added figure 1, manipulative 1, and manipulative 2 (
McAdams, David.)
20080813: Added 'More Information' and corrected step numbers in 'Finding an Inverse of a Linear Function' (
McAdams, David.)
20080405: Added color emphasis. Added understanding check (
McAdams, David.)
20070712: Initial version (
McAdams, David.)