Inverse of a Function

Pronunciation: /ɪnˈvɜrs/ ?
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.
Function
Inverse
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 boxUnderstanding 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

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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/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.

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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 drop-down 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
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. (Accessed: 2010-03-02). 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. Adam and Charles Black, 1902. (Accessed: 2010-03-02). http://www.archive.org/stream/introductiontoal00chryuoft#page/68/mode/1up/search/inverse.

Printed Resources

Cite this article as:
Inverse of a Function. 2010-03-02. All Math Words Encyclopedia. Life is a Story Problem.org. http://www.allmathwords.org/en/i/inverseofafunction.html.

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Revision History


2010-03-02: Added "References" (McAdams, David.)
2008-09-19: Added figure 1, manipulative 1, and manipulative 2 (McAdams, David.)
2008-08-13: Added 'More Information' and corrected step numbers in 'Finding an Inverse of a Linear Function' (McAdams, David.)
2008-04-05: Added color emphasis. Added understanding check (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

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