# R is the set of Reals. if F:R-> R is defined by `f(x)=x^2` for every x element R then find `f^(-1)(x)` if it exists. Find the Inverse

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### 1 Answer

We are given that `f:RR->RR` where `f(x)=x^2` and we are asked to find the inverse `f^(-1)(x)` :

**The inverse does not exist.**

(1) The graph of `f(x)` fails the horizontal line test -- e.g. the line y=4 intersects the graph at x=2 and x=-2, therefore the inverse is not a function

(2) `f(x)` is not injective (1-1 or one-to-one) as f(2)=f(-2)=4

(3) If you try to solve for the inverse, you end up with domain restrictions:

`y=x^2`

`==>x=y^2`

`y=+-sqrt(x),x>=0`

The inverse **relation** is the parabola x=y^2, which is not a function (it's graph fails the vertical line test, etc...)

The graph of the function (black) and the inverse relation (red):

**Sources:**