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.
.