Creating Inverse Function with Certain Characteristics

Question

Given the following:

• $D : \in \Re [-n, n]$
• $R: \in \Re[-\frac\pi4, \frac\pi4]$
• The curve of the function should be completely smooth, and can be undefined outside the given Domain.
• The graph should be symmetrical across $y = -x$
• $f^{-1}(f(x)) = x$ $D:\in\Re[-\frac\pi4,\frac\pi4]$ and $R:\in\Re[-n,n]$

Using this information provide both the function that meets these characteristics, and its inverse. Please show all logical steps in the derivation.

These are my thoughts:

• I figured out that based on the fact that it be completely smooth, and undefined outside of the domain, and the taking the inverse of the function results in $x$, that we are talking about an inverse trig function.
• I believe the best base function to use would be $\sin^{-1}(x)$.

I am hitting a wall at that point, and can't figure out where to go from here, considering that if $n>1$, then $\sin^{-1}(n)$ is undefined. Can someone please help?

Am I even starting on the right path?

Thanks for the help!

Answer 1

After working through this I have realized that this was actually a lot simpler then I was making it out to be, and as such I have found at least 2 solutions:

1

• given that the graph must be symmetrical about $y = -x$, and the limiting of the domain and range
• a reasonable base equation would be $\sin^{-1}(x)$
• to account for the set of Domains the variable $x\rightarrow\frac1nx \rightarrow \frac xn$
• to account for limiting the typical Range of $-\frac\pi2 < \sin^{-1}(x) < \frac\pi2 \rightarrow -\frac\pi4 < \sin^{-1}(x) < \frac\pi4$ $f(x) \rightarrow \frac12f(x)$
• final equation $f(x) = \frac12 \sin^{-1}(\frac xn)$

2 This is taking the spirit of @Daniel_Franke's suggestion:

• First consider $n = 1$ to achieve $R \in\Re[-\frac\pi4,\frac\pi4]$ with the base equation $f(x)=x \rightarrow f(x)= \frac\pi4 x \rightarrow f(x) = \frac{\pi x}4$
• Next consider unconstrained $n$ to yield $f(x)= \frac{\pi x}4 \rightarrow f(x)=\frac{\pi x}4 \frac1n \rightarrow f(x)=\frac{\pi x}{4n}$

Though this second option does not necessarily achieve the closed Domain and Range of the inverse mentioned in the final part of the givens, and it is not undefined as mentioned in the 3rd given.

As the inverses of these functions are a trivial matter I will not post them.