f(x) = x^2 + 3 and g(x) = square root x-3

What is the domain of f(g(x))?

I got f(g(x)) = x and thus the domain would be R, but this is not the answer.

Apparently I'm suppose to consider the domain of g(x) as-well? Could someone explain this please?

1 Answer

Best answer

Like you say, f(g(x)) = (sqrt(x-3))^2 + 3 = x

However, it is only x if you can actually square root (x-3) in the first place. Otherwise there is nothing to square, etc etc. So while it generally simplifies to x, it is only valid when x≥3.

Why can't I square root (x-3) in the first place?
Should I just follow the formula Dom f(g(x)) = Dom g(x) ?
What is the domain of the square root? That is when you can square root x-3... ie x is greater than/equal to 3