Consider f(t), the function describing the height of a ball thrown in the air.

Notice that for every input value t, there is exactly one output value. However, some input values have the same output. For example, f(0.5) and f(2) are both equal to 21.

What does this imply about the inverse of the function? Can you tell whether the inverse is also a function? Given a particular height, can you find the exact time the ball was at that height?

Let's graph the inverse. Recall that to graph the inverse, we must interchange the input and output values.

Which of the following graphs describes an invertible function? Select all that apply. Click "Submit" when finished.

Is the inverse a function?

Notice that there are two different times when the height of the ball is 21 feet. That is, for a given input, there is more than one output. So, the inverse is not a function.

We say that function f is not invertible.

The domain of f(t) is the set of all real numbers between 0 and 2.62.

Is it possible to restrict the domain of this function to make it invertible, that is, so each height value maps to only one time value?

Use the tool to the right to restrict the domain of f(t). Notice how restricting the domain affects the graph of the inverse.

Can you find a domain for which f(t) is invertible? Click "Done Exploring" when finished.

Restrict the domain to . Notice that the resulting graph of the inverse of f(t) is a function. So, this is one example of a domain over which f(t) is invertible.

f(t) is not invertible because there are y-values that have two corresponding x-values. However, by considering only part of the graph of f, you can eliminate the duplication of x-values.

Recall that you can tell whether a graph describes a function using the vertical line test. You can determine whether the function is invertible using the horizontal line test: If there is a horizontal line that intersects a function's graph in more than one point, then the function's inverse is not a function. If every horizontal line intersects a function's graph no more than once, then the function is invertible.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education