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Because is the distance from the center of the parabola to either side, the entire width of the water at the top will be The trough is 3 feet (36 inches) long, so the surface area will then be:
This example illustrates two important points:
Functions involving roots are often called radical functions . While it is not possible to find an inverse of most polynomial functions, some basic polynomials do have inverses. Such functions are called invertible functions , and we use the notation
Warning: is not the same as the reciprocal of the function This use of “–1” is reserved to denote inverse functions. To denote the reciprocal of a function we would need to write
An important relationship between inverse functions is that they “undo” each other. If is the inverse of a function then is the inverse of the function In other words, whatever the function does to undoes it—and vice-versa.
and
Note that the inverse switches the domain and range of the original function.
Two functions, and are inverses of one another if for all in the domain of and
Given a polynomial function, find the inverse of the function by restricting the domain in such a way that the new function is one-to-one.
Show that and are inverses, for .
We must show that and
Therefore, and are inverses.
Find the inverse of the function
This is a transformation of the basic cubic toolkit function, and based on our knowledge of that function, we know it is one-to-one. Solving for the inverse by solving for
So far, we have been able to find the inverse functions of cubic functions without having to restrict their domains. However, as we know, not all cubic polynomials are one-to-one. Some functions that are not one-to-one may have their domain restricted so that they are one-to-one, but only over that domain. The function over the restricted domain would then have an inverse function . Since quadratic functions are not one-to-one, we must restrict their domain in order to find their inverses.
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