ON INVERSE FUNCTIONS

With Domain Restrictions

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 1: Find the inverse of the function .

Solution:

By now you should recognize that this is the equation of a parabola. Since a parabola is "U" shaped, it is not one-to-one. Take a look at the graph below. We know that if we divide the x-axis into two parts at the point x = 8, the graph on the left side of the line x = 8 is one-to-one, and the graph on the right side of the line x = 8 is also one-to-one. The red part of the x-axis is the set of all real numbers in the interval , and the blue part of the x-axis is the set of all real numbers in the interval

You could have also discovered the x = 8 dividing point by finding the vertex of the parabola. You can find the vertex of parabola by rewriting the original equation so that we have . From the last form, we can read the vertex as (8, -44). We are only interested in the x-part of the point, the 8.

and set it equal to x.

because the range of the inverse is equal to the restricted domain of the original function.

Check:

Suppose you restricted the domain . Choose a number from this domain, say 10. The function f takes the 10 to - 40 because . The inverse function takes the - 40 to 10 because .

Suppose you restricted the domain to . Choose a number from this domain, say 0. The function f takes the 0 to 20 because . If the inverse function, , takes the 20 back to the 0, we have correctly worked the problem. The inverse function takes the 20 back to 0 because

Review another problem concerning of finding the inverse of a function where the domain of the original function needs to be restricted.

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Author:Nancy Marcus

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