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 or
Notice this new graph is a reflection of the original graph over the line y = x.

Check your understanding: (Remember to write down your answers.)

1.  Find the inverse for the functions:

a. 

b.  y = 4(x - 2)

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

a. 

b. 

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rev.  01/06/2007