Inverting Non-Invertible Functions - Maple Help

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Inverting Non-Invertible Functions

Main Concept

Table 1: Examples of Inverse Functions

 Function Inverse Function Function Inverse Function Function Inverse Function $\mathrm{sin}\left(x\right)$ $\mathrm{arcsin}\left(x\right)$ $\mathrm{cos}\left(x\right)$ $\mathrm{arccos}\left(x\right)$ $\mathrm{tan}\left(x\right)$ $\mathrm{arctan}\left(x\right)$ $\mathrm{csc}\left(x\right)$ $\mathrm{arccsc}\left(x\right)$ $\mathrm{sec}\left(x\right)$ $\mathrm{arcsec}\left(x\right)$ $\mathrm{cot}\left(x\right)$ $\mathrm{arccot}\left(x\right)$ ${x}^{n}$ $\sqrt[n]{x}$ ${ⅇ}^{x}$ $\mathrm{ln}\left(x\right)$ $W\left(x\right)$

Inverting Non-Invertible Functions

There is clearly a problem with the "inverses" shown in the table of Examples of Inverse Functions, because many of the functions listed in this table are not one-to-one, and hence are not invertible. For example, $\mathrm{sin}\left(0\right)=0=\mathrm{sin}\left(\mathrm{π}\right)$, , and so on. To handle this situation, it is customary to restrict the domain of the original function down to a subdomain on which the function is invertible. This subdomain should be large enough that all possible values of the original function are obtained by the function restricted to this subdomain. That is, the range of the function restricted to this subdomain should be the same as the range of the original function. For the function $\mathrm{sin}\left(x\right)$, it is normal to restrict to the interval $\left[-\frac{\mathrm{π}}{2},\frac{\mathrm{π}}{2}\right]$. These graphs show the $\mathrm{sin}\left(x\right)$ function on this restricted domain and its inverse function, $\mathrm{arcsin}\left(x\right)$:

 

Notice that the range of the $\mathrm{arcsin}\left(x\right)$ function is $\left[-\frac{\mathrm{π}}{2},\frac{\mathrm{π}}{2}\right]$, which is the domain of the (restricted) $\mathrm{sin}\left(x\right)$ function.

Some authors also use the notation ${\mathrm{sin}}^{-1}\left(x\right)$ for the arcsin function, and similarly for the other trigonometric functions. This notation is more in keeping with the generic ${f}^{-1}\left(x\right)$ notation. It is important to remember that ${\mathrm{sin}}^{-1}\left(x\right)=\mathrm{arcsin}\left(x\right)$, not $\frac{1}{\mathrm{sin}\left(x\right)}$.

In the demonstration below, choose a function in the Select function list. The graph of the function is shown in the first plot.

Click Show Invertible Restriction and the standard restriction on the graph will be highlighted in the second plot.

Finally, click Invert Restricted Function to show the result of inverting the function in the last plot.

Try selecting different functions.

 Select function: xsin(x)cos(x)tan(x)arcsin(x)arccos(x)arctan(x)e^x10^xln(x)1/xx^2x^3x+sin(x)x*e^x

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