The functions $f$ and $g$ have domains $(0,\infty)$ and $(2,\infty)$ respectively and are defined by
$$f(x)=x^2-1$$
$$g(x)=2x-1$$
Find the domain and range of $fg(x)$
The question for this specific problem was find the domain and range for $fg(x)$, however i keep getting the wrong answer the way i look at it is that the domain of $g(x)$ leads to values of $(3,\infty)$ however the $f(x)$ function can only take values between $(0,\infty)$ therefore it need to make $g(x)=0$ then from there find the values for which it can take in which case i get the domain for the composite function to be $(0.5,\infty)$ however the text book states that the answer is $(2, ∞)$ any help is much appreciated question Pt.1 question Pt.2
The domain of $f\circ g$ is the set on which it is defined, and it is defined for all $x>2$. Suppose that $x>2$. Then $x$ is in the domain of $g$, so $g(x)$ is defined. Moreover, as you said, $g(x)>3$, so in particular $g(x)>0$. Thus, $g(x)$ is in the domain of $f$, and $f(g(x))$ is therefore defined. The fact that some elements of the domain of $f$ aren’t in the range of $g$ is irrelevant.
To complete the argument, we note that if $x\le 2$, then $g(x)$ isn’t even defined, so $f(g(x))$ certainly isn’t defined. Thus, $x$ is in the domain of $f\circ g$ if and only if $x>2$.