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Is there an algebraic(without graphs) way to determine the existence of a function's inverse without using calculus?

I'm an undergrad engineer and can obviously solve this using basic calculus, but I'm having trouble finding a way to explain how to do this using only 11th grade math (explaining it to my sister).

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1 Answer

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Write $f(x)=y$. Try to solve for $x$. If the algebra gives a unique solution then that solution is precisely $f^{-1}(y)$.

For example, $f(x)=3x-1$ solve $y=3x-1$ to get $x = (y+1)/3$ hence $f^{-1}(y)=(y+1)/3$.

On the other hand, $f(x)=x^2$ solve $y=x^2$ to get $x = \pm \sqrt{y}$. Oops. Two values, not a function. In order for an inverse to exist for this $f$ we would have to cut down the domain.

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