5.7 Inverses and radical functions  (Page 2/7)

Because $x$ is the distance from the center of the parabola to either side, the entire width of the water at the top will be $2x.$ The trough is 3 feet (36 inches) long, so the surface area will then be:

This example illustrates two important points:

1. When finding the inverse of a quadratic, we have to limit ourselves to a domain on which the function is one-to-one.
2. The inverse of a quadratic function is a square root function. Both are toolkit functions and different types of power functions.

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 $f^{-1}(x).$

Warning: $f^{-1}(x)$ is not the same as the reciprocal of the function $f\left(x\right).$ This use of “–1” is reserved to denote inverse functions. To denote the reciprocal of a function $f\left(x\right),$ we would need to write ${\left(f\left(x\right)\right)}^{-1}=\frac{1}{f\left(x\right)}.$

An important relationship between inverse functions is that they “undo” each other. If $f^{-1}$ is the inverse of a function $f,$ then $f$ is the inverse of the function $f^{-1}.$ In other words, whatever the function $f$ does to $x,$ $f^{-1}$ undoes it—and vice-versa.

and

Note that the inverse switches the domain and range of the original function.

Restricting the domain to find the inverse of a polynomial function

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.