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Learning intentions:
In this section we will examine:
- What an inverse function is and when a function will have an inverse function.
- How to fully define an inverse function.
- How to graph the inverse function.
- How to determine the point of intersection between a function and its' inverse.
The inverse of a function
In mathematics you have already encountered inverse operations such as addition and subtraction or multiplication and addition. In both of these examples, one operation 'undoes' the other. We can define function and their inverse functions in a similar way.
Defining the inverse of a function
Let f be a one-to-one function with a domain, dom( f ), and range, ran( f ). Then the inverse function, denoted f -1, is defined by:
Remember: A relation is a function when every x-value has, at most, one corresponding y-value.
Note: A function will only have an inverse function if it is one-to-one, all other functions will have inverse relations.
That is, the inverse function evaluated at y will return the corresponding x-value which gives y in the original function.
The domain and range of the inverse function
For the one-to-one function function, f, with a domain, dom( f ), and range, dom( f ). The inverse function, f -1, will have:
- dom( f -1) = ran( f )
- ran( f -1) = dom( f )
Geometric interpretation of the inverse function
Geometrically, the inverse function is a reflection of the function over the line y = x. This is because the inverse is found by swapping x and y in the equation (method illustrated below).
The dynamic GeoGebra worksheet illustrates the trace of the inverse function of f (x) = x2.
- Please click on the play button in the bottom left hand corner to animate!
Method: Finding the inverse of a function
To find the inverse of a function, f, use the following method:
- Write the equation in the form y = f(x).
- Swap all terms of x and y for the other to find the inverse function.
- Solve the inverse equation for y.
- Express the inverse function using correct notation, f -1, and specify the domain.
Remember: The domain of the inverse function will be equal to the range of the original.
Finding the intersection between a function and its' inverse
To find the point of intersection between a function, f, and its' inverse function, f -1, you can solve any of the following three equations.
All three equations will find the intersection because the intersection, if it exists, will lay on the line y = x. If you are solving by-hand, make sure you choose the simplest equation!
7E - VIDEO EXAMPLE 1:
Fully define the inverse of the function below:
7E - VIDEO EXAMPLE 2:
Fully define the inverse of the function below:
Success criteria:
You will be successful if you can:
- Find the inverse of a function and express it using correct notation.
- Graph the inverse of a function and find its' intersection with the original function.
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