Determining & Proving Inverse Functions: Methodology?

[jgens]

Does anyone know of an existing methodology for determining and proving whether or not a function has an inverse?

Thanks.

 

[nicksauce]

How about by determining if the function is a bijection? Or am I missing something here?

 

[HallsofIvy]

The "horizontal line test". How many times does any horizontal line cross the graph of the function? Which is the same as "determining if the function is a bijection"!

 

[jgens]

Thank you very much. Just a brief follow up question: Suppose a function is invertable but the inverse has no representation in terms of elementary functions (I'm including trigonometric and logarithmic functions in this category) - I think f(x) = x + sin(x) would meet this criterion - is there a general method for proving that the function does not have an inverse expressable in terms of elementary functions?

Sorry if that doesn't make sense.