## (Section 12)

During this year you will need to use the idea of inverse functions, so let's review some of the basic concepts you should remember.

• How is an inverse function formed?

An inverse function is formed by exchanging x for y.

If y  = 3x + 2, then the inverse function would be x = 3y + 2 would be the inverse function.  But you can solve this equation if you would like for .

• When does a function have an inverse function?

A function must pass the horizontal line test to have an inverse function.  This is necessary because the definition of a function requires that for every x there exists exactly one y.  Remember the same y value can be assigned to more than one x value in a function.

If an original function has two x values which yield the same y value, then when the x and y values are switched there will be two y values for the same x value.  This would prevent the switched equation from being a function.  Therefore, it is necessary to restrict the domain on some functions so that they will pass the horizontal line test.

 Suppose .  Recall that this function does not pass the horizontal line test.  x=1 and x = -1 both yield the same y value of 1. It is necessary to restrict the domain of the function so that it passes the horizontal line test.  You have two choices.  Either define it as or .  I have selected the first choice. Now when the x and y are switched the equation becomes Notice this new graph is a reflection of the original graph over the line y = x.

1.  Find the inverse for the functions:

a.

b.

2.  Find the inverse for each of the following functions after restricting the domain so the function passes the horizontal line test.

a.

b.