Would I be correct in assuming that the domain of the composite function $(f∘g)(x)$ is the intersection of the range of $g$ and the domain of $x$? If not, how do I find the domain of a composite function?
Would I find the range of $(f∘g)$ as I would find the range of a non-composite function, or do I need to use the components $g$ and $f$?
P.S.: Are $D_f$ and $R_f$ accepted notations for the domain and range of $f$ respectively?
Best Answer
NOTE: I'm going to use the more precise term "image" instead of "range" because I've heard people use range to mean either image OR codomain and I'd prefer to be unambiguous.
Given two functions $f: X \to Y$ and $g: Y \to Z$, the composition $g\circ f$ has domain $X$, codomain $Z$, and the image (range) is given by the image of the restriction of $g$ to the image of $f$.
Example: Let $f: \Bbb R \to \Bbb R$ be given by $f(x)=x^2$ and let $g: \Bbb R \to \Bbb R$ be given by $g(x) = 2x+3$. Then the composition $g\circ f$ is given by $(g\circ f)(x)=g(f(x)) = g(x^2) = 2x^2+3$. Because the domain of $f$ is $\Bbb R$, the domain of $g\circ f$ is $\Bbb R$. Because the codomain of $g$ is $\Bbb R$, the codomain of $g\circ f$ is $\Bbb R$. With that we can write down the signature of the function as $g\circ f: \Bbb R \to \Bbb R$.
But the image of the function takes a little thought to figure out. We need to find the image of the restriction of $g$ to the image of $f$. So first, what's the image of $f$? We can see that it is the nonnegative real numbers $\Bbb R_{\ge 0}$. So then what is the image of the function $\left.g\right|_{\operatorname{Im}(f)}: \Bbb R_{\ge 0} \to \Bbb R$? I'm not sure how to show it without some calculus, but hopefully it's pretty clear that it's the interval $[3, \infty)$. So that is the image of $g\circ f$.
So, to sum up: $$\begin{align}\operatorname{Dom}(g\circ f) &= \Bbb R \\ \operatorname{Codom}(g\circ f) &= \Bbb R \\ \operatorname{Im}(g\circ f) &= [3,\infty)\end{align}$$
As to your question about notation, I haven't seen that notation before (for whatever that's worth), but as long as you define it somewhere on whatever paper/ homework you're using it in, it should be fine.