f(x)=sqrt(x-2) what is the inverse of f?

Question

I got inverse f(x)= x^2+2 however, you need to specify the domain of x.

the answer key says when x=>0
can you please explain why? I thought it would be when x=>2 ?

Answer 1

domain of f is x >= 2
Note that the range of f inverse is also f >= 2, since x^2 >= 0 for all x.

Since f(x) = √(x-2), f >= 0
because √4 = 2, not ±2!

So, since the range of f is f >= 0, the domain of f inverse is also x >= 0.

f inverse is not 1-1, so you have to take only the branch whose inverse is f(x), namely, where x>=0.