Composite functions

Question

How would you describe the existence of a composite function $f(g(x))$in terms of range of $g$ and domain of $f$ . Does range of $g$ need to be subset of domain of $f$ or is it sufficient if the two sets have intersection only?
I used to define composite function if range of $g$ is a subset of domain of $f$ and in that case domain of the composite function is same as domain of the function $g$ but I have come across questions where only intersection exists. In that type of situation domain of composite function can not be the domain of $g$. I have this issue related to high school mathematics. Could you please help me to find the most appropriate way of describing the issue to the students relevant to their level of studies .
For further clarification here you have examples

1. Let $f(x) = x^2$ and $g(x) = \sqrt{x-1}$.
   Here range of $g$ is a subset of domain of $f $, therefore $f(g(x)) = x -1$ exists. Now the issue is can you say $f(g(0))= - 1$ because 0 is not in the domain of $g$?

2. Let $f(x)= x + 2$ and $g(x) = 1/(x - 1)$.
   Here range of $f$ is not a subset of domain of $g$ , now what about $g(f(x)) = 1/(x+1)$, how can you explain the way to obtain domain of this composite function because here that is not the same domain of $f$?

3. Do we need to treat finding expression for $g(f(x))$ and finding composite function $g(f(x))$ in two different ways?
   Edited
   By going through the suggested answers following are the conclusions I could able to make ,
   

1. When the terms function or domain not mention in the question we can treat$ f(g(x))$ as an expression and substitute any real value to $x$ if output is real.
   
2. When the domain of $f$ and $g$ are given $f(g(x))$ can be defined if range of $g$ is a subset of domain of $f$.
   
3. When domain of $f$ and $g$ are not given we have to determine the domain of $g$ such that range of $g$ is a subset of domain of $f$ and range of $f(g(x))$ should be determined according to the selected domain of $g$.
   If you have any exceptions please mention it in your answers or comment about it so that we can make the final conclusion.

Answer 1

Before addressing some of the issues directly, let me mention what I think is the standard approach in mathematics. A function is typically defined together with its domain and codomain, so saying "function $f$" is a shorthand for "function $f \colon A \to B$" (where $B$ is the codomain and potentially the range $f(A)$ is a proper subset of $B$). And then, the standard way of introducing composition is to require two functions $f \colon A \to B$, $g \colon B \to C$ to have matching (co)domains and let $g \circ f \colon A \to C$, $(g \circ f) (x) = g(f(x))$.

Of course, this is not the only option. If $f \colon A \to B$ and $g \colon B' \to C$ don't have matching (co)domains but $B' \subseteq B$ (typically $B = \mathbb{R}$), you can still define $g \circ f$ on the set $A' := f^{-1}(B')$ by $(g \circ f) \colon A' \to C$, $(g \circ f)(x) = g(f(x))$. It's mostly a pedagogical choice. If one wanted to be pedantic and stick to the previous definition, one would just have to use the composition $g \circ f|_{A'}$ (with $f$ replaced by its restriction).

It's probably best to stick with whatever your primary reference for the subject uses.

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Now, there's a type of problems where a function is only given by an expression (say, $\frac{1}{x}$) and the student is required to find the set of all $x$'s for which the expression makes sense, the so called natural domain (here it would be $\mathbb{R} \setminus \{0\}$).

However, it's important to make a distinction between a function and an expression. Formally, in problems like the one above you are given an expression and asked to make it into a function. I'd suspect not all students are ready for such subtleties, but there are some advantages:

• This is how it works in mathematics, but even more importantly, in programming. There, functions are usually statically typed, meaning that you specify the type of their input and output. If you try to compose two functions: $$ f \colon \mathbb{N} \to \mathbb{R}, \ f(n)=\sqrt{n} \quad \text{with} \quad g \colon \mathbb{N} \to \mathbb{N}, \ g(x)=x^2, $$ in a typical programming language, you should expect an error when trying to compute $g(f(4))$, even though the reason in somewhat artificial (at least from the mathematical perspective).
• The distinction can clear up some confusion in your example 1. If $f(x) = x^2$ and $g(x) = \sqrt{x-1}$, then actually we deal with functions $g \colon [1,\infty) \to \mathbb{R}$ and $f \colon \mathbb{R} \to \mathbb{R}$, and their composition is by definition $f \circ g \colon [1,\infty) \to \mathbb{R}$, $(f \circ g)(x) = f(g(x)) = x-1$. Of course, given the expression $x-1$ out of context, we would ascribe $\mathbb{R}$ as its natural domain. Avoiding unnecessary subtleties, one could also summarize it as follows: the natural domain of the composition $f \circ g$ may be larger than the natural domain of $g$.

I hope it also sheds some light on how you could explain points 2 and 3.

Answer 2

I initially found requiring the range of $g$ being a subset of the domain of $f$ to be the more appealing option, but having thought about it more, I now think that any two real functions should be composable. This is based on two principles:

• The key takeaway skill from talking about domains is that students should be able to recognize when an expression exists or doesn't exist for various $x$. There shouldn't be extra rules for dealing with functions for students to memorize. As such:
• Functions should not behave differently from ordinary algebra: students can write algebraic expressions that don't exist for any $x$, so they should also be able to write $f(g(x))$ when it doesn't make sense for any $x$.

Of course, the most important principle is to agree with whatever your textbook says, so that students don't get confused when they reference it.

Answer 3

I think the answer by Michal Miśkiewicz does an excellent job addressing issues, and captures the formal mathematical perspective.

The distinction between the formality of "function with domain" and the more laissez-faire perspective of "expressions" is an important pedagogical issue. When I teach composition in precalculus and come to this issue, I like to refer to the "function with domain" as a rule typically given by an algebraic expression along with a demon who prevents evaluation at values not in the domain. This is akin to Maxwell's demon, the proverbial gatekeeper of the Second Law of Thermodynamics.

Some of my more artistic students enjoy drawing their creative renditions of gatekeepers of functions domains. Often the characteristics of the demon reflect the name of the function--"square root" inspires some to draw a demon holding a mandrake with square roots. Having students draw domain demons so I can share them with the class seems to help students retain the idea that functions have domains. When I grade exams, I sometimes see more sketches that students draw in the margins.

When the demon goes to sleep, the function becomes an expression and that's when we can play fast and loose. When the demon for $(f(x)=x^2,\ x\in {\bf R})$ goes to sleep, we can square all sorts of things such as matrices, mandrake roots, etc.