How do I solve this function and find its domain?

Question

Suppose that $f(x)= -x^2+1$ and $g(x)= \sqrt{x}$. How do we find $f \circ g$ and $g \circ f$ and their domains?

Answer 1

Hint: For the first one, recall that

$$(f \circ g)(x) = f(g(x))$$

So $x$ has to be in the domain of $g$, and $g(x)$ has to be in the domain of $f$. The domain of the square root function is $[0, \infty)$, but $f$'s domain is everything.

Now to find the function, just evaluate $f(g(x))$:

$$f(g(x)) = -(g(x))^2 + 1 - -(\sqrt{x})^2 + 1 = -x + 1$$