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I tried to find the inverse of a function on wolfram's inverse calculator, but when I hit enter it said: "No result found in terms of standard mathematical functions," hence my question above.

I am well aware of indefinite integrals that have no result in terms of standard mathematical functions, and have to be numerically approximated, such as the normal distribution curve used in statistics. I am however, not aware of any inverse relations that have no result in terms of standard mathematical functions.

John Zimmerman
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1 Answers1

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So long as your function is one-to-one, there will be an inverse function from the range back to the codomain. We have to decide what we mean by standard function, but if we take the basic functions from calculus, there are simply too many ways that we can combine them. Using standard operations of $+, -, \times, \div$ it is only in the rarest cases that there will be a way to solve and get a nice expression.

For an example, take the sum of any two of the increasing functions $e^x$, $\ln(x)$, and $\arctan(x)$. These will be increasing and hence one-to-one, but there won't be a 'standard' way to express the inverse, even though it exists.

If you go out of the most common comfort zone, you can sometimes find notation for a few combinations. The product logarithm for the function $f(x)=xe^x$ is one such example. Remember that we can always assign a name to a function that didn't have one before and toss it into our toolkit with the other more famous ones.

user1390
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