Inverse of a function

A function has to be bijective in order to have an inverse.

 

A function can be locally bijective, so it's inverse exists only in some finite interval.

For example [tex]x^{2}[/tex] is not a bijective in any interval containing x=0 (since f'(0)=0) but if you restrict yourself to x>0, then you off course have the inverse
[tex]f(x)=\sqrt{x}[/tex] or in x<0 the inverse is [tex]f(x)=-\sqrt{x}[/tex].

 

Another example. f(x)= e^x, as a function from R to R, is not surjective, so not and does not have an inverse. In particular, there is no f^-1(-1). But if I consider it as a function from R to R⁺, the positive real numbers, then it is bijective and f^-1(x)= ln(x).

A function, from A to B, has an inverse if and only if it is bijective.

 

I had a little confusion in defining the inverse sorry.

For [tex]x\geq 0[/tex] the inverse of [tex]y=x^{2}[/tex] is [tex]x=f^{-1}(y)=\sqrt{y}[/tex]


For [tex]x\leq 0[/tex] the inverse of [tex]y=x^{2}[/tex] is [tex]x=f^{-1}(y)=-\sqrt{y}[/tex]

Notice of course that the inverse is defined only over [tex]y\geq 0[/tex], since the range of x^2 is only the non-negative real numbers.

Also, the interval where the function can be define as a bijection may inclue x=0, but only as a boundary point.