ON INVERSE FUNCTIONS
With Restricted Domains
You can always find the inverse of a one-to-one function without restricting the domain of the function. Recall that a function is a rule that links an element in the domain to just one number in the range. You could have points (3, 7), (8, 7) and (14,7) on the graph of a function. A one-to-one function adds the requirement that each element in the range is linked to just one number in the domain. In this case, the above three points would not be points on the graph of a one-to-one function because 7 links to different numbers in the domain.
You can identify a one-to-one function from its graph by using the Horizontal Line Test. Mentally scan the graph with a horizontal line; if the line intersects the graph in more than one place, it is not the graph of a one-to-one function. If the function is not one-to-one, then its inverse will not be unique, and the inverse function must be unique. The domain of the original function must be restricted so that its inverse will be unique.
This section will show you how to restrict the domain and then find a unique inverse on that domain. Study the graph of a function that is not one-to-one and choose a part of the graph that is one-to-one. We will find the inverse for just that part of the graph.
The following is an example of finding the inverse of a function that is not one-to-one.
Problem 3: Find the inverse of the function .
You could have also discovered the dividing point by finding the vertex of the parabola. You can find the vertex of parabola by completing the square of the original equation. If we can get the original equation into the form , we know the vertex is .
Start by rewriting the original equation so that we have .
Factor from the first two terms and rewrite as .
Add 0 inside the parenthesis in the form .
Keep in mind that each of the forms of the original equations are all equivalent. This means that if we evaluate each of these equations by the same number, all the results will be the same.
We are going to change the looks again by expanding the above equation to.
Factor from the first three terms and simplify the rest.
This can be simplified to . The vertex is .
We are only interested in the x-part of the vertex, the . The line divides the graph into the graphs of two one-to-one functions, one on either side of the line..
Review another example of finding the inverse of a function where the domain of the original function needs to be restricted.
Work the following problems and check the answers and solutions.
1. Find the inverse of the function . Answer.
2. Find the inverse of the function . Answer.
3. Find the inverse of the function . Answer.
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