Question on finding an inverse function (dealing with signs)

Question

Find the inverse of the function $f(x)=25-x^2$ edit: when $x \geq 0$

So, I went $y=25-x^2$ $\rightarrow x=25-y^2$

Then, solving for $y$ I get $y = \pm \sqrt{25-x}$

I'm just wondering if I use the $+$ or the $-$ and why. Thanks

Answer 1

We have $$y=25-x^2$$ solving this for $x^2$ we get $$x^2=25-y$$ so $$|x|=\sqrt{25-y}$$ and in the case of $x\geq 0$ we get $x=\sqrt{25-y}$ in the other case $x=-\sqrt{25-y}$

Answer 2

The function is not injective (one-to-one), so no global inverse exists. You can speak of local inverses for the intervals $\left(-\infty, 0\right]$ and $\left[0,\infty\right)$, however.

Answer 3

If you take both the positive and negative, you get the curve that is inverse to $f(x)$. That whole curve $\pm \sqrt{25-x}$ is the inverse and has a positive and negative branch, but we must emphasize it's a curve not a function