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Inverse Functions

If tex2html_wrap_inline86 is 1-1 and onto then the correspondence that goes backwards from Y to X is also a function and is called f inverse, denoted tex2html_wrap_inline216 . As this is the symbol used for inverse image of a set it must be clear from the context how the symbol is being used. This map is easily described by tex2html_wrap_inline218 and tex2html_wrap_inline220 if and only if y=f(x). So if f(3)=5 then tex2html_wrap_inline226 automatically. If tex2html_wrap_inline228 then tex2html_wrap_inline230 . In terms of pairs this says if (17,-8) is in the graph of f then (-8,17) is in the graph of tex2html_wrap_inline216 .

This relationship is easy to remember for real functions since switching coordinates of a point in the plane puts you at the reflection of the original point about the line y=x. Thus the graph of tex2html_wrap_inline216 must be the reflection of the graph of f about the line y=x. This is a great help if the graph of f is already known.  

It's the 1-1 condition that is really critical for constructing an inverse function. If f is 1-1 but not onto we can simply replace the codomain with the range f(X) so that tex2html_wrap_inline254 is then 1-1 and onto so we can talk about an inverse tex2html_wrap_inline256 .

There are some obvious questions that arise here. What are the proper domain and range for an inverse function?  Given a formula for f, how does one find a formula for tex2html_wrap_inline216?  Can a formula always be found? 

If a function is claimed to be the inverse of a given function, this claim can be checked by a pair of well-known formulas that tell exactly how f and tex2html_wrap_inline216 fit together under composition. For tex2html_wrap_inline86 1-1 and onto we always have:


If tex2html_wrap_inline268 looks strange at first, keep in mind we used tex2html_wrap_inline60 and tex2html_wrap_inline80 earlier only to emphasize the fact that we had two elements that came from two possibly different sets. In general, we can call elements of a set by any name we choose. Here we have two separate equations, so it's all right to use tex2html_wrap_inline60 in one of them and tex2html_wrap_inline268 in the other. It is easy to verify these properties. For the first one, start with letting tex2html_wrap_inline278 . We would like to show that z=x. Since tex2ht
ml_wrap_inline282 is equivalent to f(x)=f(z) by the definition of inverse, we are done as the 1-1 property says then that z=x. The second property has the same proof.

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Dan Rinne
Thu Aug 8 16:22:12 PDT 1996