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 .

**Solution:**

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..

- The identify function is . The graph of the identify function is the straight line that bisects the first and third quadrants and passes through the origin. All the points on the graph of this line have an x- and y- coordinate that are equal, for example, points similar to (2, 2) and (7, 7) would be found on the graph of this line.
- Since an inverse function is a kind of "UNDO" function, the composition of a function with its inverse is the identify function. For example, if the rule f(x) takes a 3 to 10 and the inverse function takes the 10 back to the 3, the end results is that the composite of the two functions took 3 to 3. This is the identify function.
- We know then that . Let’s use the rule for the f(x) function with the argument

- Subtract

- Multiply both sides of the equation by

- Add

- Factor the left side of the equation.

- Take the square root of both sides of the equation.

- Add

**Now we have a problem.**We have come up with two inverses and inverses must be unique. Which one shall we choose as the inverse, or ? It depends on what you choose as your restricted domain.- If you choose the restricted domain to be , the inverse is because the range of the inverse is equal to the restricted domain of the original function.
- If you choose the restricted domain to be , the inverse is because the range of the inverse is equal to the restricted domain of the original function.
- Does it matter which one you choose? Not for this problem. However, if you were dealing with a real-life problem, it could. For example, if you were interested in the length of fish, you would not want negative weights and thus would choose the domain

**Check:**

**Check:**

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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