That depends on how you define an inverse. [imath] y=x^2 [/imath] has no inverse function and we can say it has no inverse.
An inverse is a unary operation. It requires to know on which set!
If we use the language (set) of relations then inverting a relation is simply interchanging the components. If there is a relation [imath] R [/imath] between the sets [imath] A [/imath] and [imath] B [/imath], sometimes written [imath] ARB [/imath] or [imath] A\sim_R B [/imath] then [imath] R [/imath] is a subset of [imath] A\times B .[/imath] Say we have [imath] R=A\times B [/imath] to keep it simple. This means [imath] R=\{(x,y)\,|\,x\in A, y\in B\} .[/imath] The inverse relation would then be [imath] R^{-1}=\{(y,x)\,|\,x\in A,y\in B\} .[/imath] Since [imath] f [/imath] is a relation, we can define a relation [imath] f^{-1} [/imath] by [imath] f^{-1}=\{(f(x),x)\} [/imath] instead of [imath] f=\{(x,f(x))\} .[/imath]
If [imath] f(x)=x^2 [/imath] then [imath] (-1,1),(1,1)\in f [/imath] and [imath] (1,-1),(1,1) \in f^{-1} .[/imath] However, this isn't a function any longer, since this would mean that we have [imath] f^{-1}(1)=-1 [/imath] and [imath] f^{-1}(1)=1 .[/imath]
Long story short ...
... which is no function.
