Composing when Functions are Sets of Points

What is the notation for composition of functions?

The notation for composition of functions is an open circle, kind of like a "degree" symbol, but a little bigger and more in the middle of the line's height. You can see the difference here:

When you write the "composition" symbol, make sure that the circle is open, not closed (that is, no filled in).

The filled-in (and somewhat smaller) dot indicates multiplication, which is totally *not* what we're doing here.

In addition to the open-circle notation for function composition, there is also the more intuitive notation, where the one function is explicitly plugged into the other one:

The open-circle notation is read as "eff compose gee of eks"; the intuitive notation is read as "eff of gee of eks".

What is the difference between the composition of g "and" f and the composition of g "with" f?

There is no difference between "the composition of g and f" and "the composition of g with f". Each means the same thing; each expression can be written in any of the following equivalent ways:

• g(f(x))
• (g ∘ f)(x)
• g(x) ∘ f(x)

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Let's start simple, by composing functions that are defined not by formulas or rules, but by sets of (x, y) points.

Can a function be a set of points?

A function can be a set of points. Actually, all functions are sets of points. It's just that these sets are almost always written in terms of the rules that associate the x's and the y's. Nearly all functions have infinitely many points, so we use the rule instead of a never-ending list of points. As long as they have output y-values for every input value and no input x-value is repeated, then, yes, a set of points is indeed a function.

How do you do function composition with ordered pairs?

Suppose we have functions f and g, where each function is defined by a set of (x, y) points. To do the composition g(f(x))), we follow these steps:

1. Choose a point in the set for f.
2. Take the x-value of that point as the input into f.
3. The output of f is the y-value from that same point.
4. Find the point in the set for g that has the same value for its x-value as the y-value from f. (If there is no such point, then the composition fails; it is "not well defined".)
5. Plug this value into g.
6. The output of g is the y-value from that same point.

As an example, let's suppose we have these two functions:

Then the composition f(g(2)) starts by finding the point in the set for g that has 2 as its x-value. This point is (2, 1).

The output (or y-value) of this point in g is 1; g(2) = 1.

We're plugging g into f, so we need to find a point in f that has 1 as its x-value. This point is (1, 3).

The output (or y-value) of this point in f is 3; f(g(2)) = f(1) = 3.

This tracing of values from x-value to y-value to x-value, etc, etc, is all there is to composing functions that are (finite) sets of points.

What is an example of composing with points?

• Let f = {(−2, 3), (−1, 1), (0, 0), (1, −1), (2, −3)}
  and let g = {(−3, 1), (−1, −2), (0, 2), (2, 2), (3, 1)}.
  Find the following:
1. f (1)
2. g(−1)
3. (g ∘ f )(1)

Note that they never told us what were the formulas, if any, for f(x) or g(x); we were only given a list of points for each. But these lists are sufficient for answering the question, as long as we keep careful track of our x- and y-values.

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• Let f = {(−2, 3), (−1, 1), (0, 0), (1, −1), (2, −3)} and
  let g = {(−3, 1), (−1, −2), (0, 2), (2, 2), (3, 1)}.
  Find the following:
1. (f ∘ g)(0)
2. ( f ∘ g)(−1)
3. (g ∘ f )(−1)

Part (iii) of the above example points out an important consideration regarding domains and ranges. It may be that your composed function (the result you get after composing two other functions) will have a restricted domain, or at least a domain that is more restricted than you might otherwise have expected. This will be more important when we deal with composing functions symbolically later on.

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Another exercise of this type gives you two graphs, rather than two sets of points, and has you read the points (that is, the functions' values) from these graphs.

What is an example of composing functions from their graphs?

• Given f (x) and g(x) as shown below, find (f ∘ g)(−1).

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• Given f(x) and g(x) as shown in the graphs below, find (g ∘ f )(x) for integral values of x on the interval −3 ≤ x ≤ 3.

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You aren't generally given functions as sets of points or as graphs. However, you should probably expect one question of this type on the next test, as a check of your conceptual understanding.