Determining & Proving Inverse Functions: Methodology?

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To determine if a function has an inverse, one can check if it is a bijection, which can be visually assessed using the horizontal line test. If any horizontal line intersects the graph of the function more than once, the function does not have an inverse. Additionally, the discussion raises the issue of functions that are invertible but whose inverses cannot be expressed in terms of elementary functions, such as f(x) = x + sin(x). There is a query about whether a general method exists to prove that such functions lack an inverse in elementary terms. Understanding these concepts is crucial for analyzing the invertibility of functions.
jgens
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Does anyone know of an existing methodology for determining and proving whether or not a function has an inverse?

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

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