Math Revision 16.4 - Composite Functions

In these revision notes for Composite Functions, we cover the following key points:

• What is a composite function?
• How to express a composite function?
• How to combine two or more functions to obtain a single composite function?
• What are the properties of composite functions?
• How to evaluate a composite function?
• What happens when a function is composed of itself?
• How to express a composite function schematically?
• How to evaluate composite functions from graphs?
• How to evaluate composite functions from tables?
• How to find the domain of a composite function?
• How to find the range of a composite function?

Composite Functions Revision Notes

By definition, a composite function is a new function obtained when one function is used as the input value for another function. Essentially, the output of the inner function (the function used as the input value) becomes the input of the outer function (the resulting value).

If f(x) acts the input of another function g(x), then we express the resulting composite function as g[f(x)] or g ◦ f(x). We read it as "the function g of f(x)" or "the function g around f(x)".

Likewise, if g(x) acts the input of another function f(x), then we express the resulting composite function as f[g(x)] or f ◦ g(x). We read it as "the function f of g(x)" or "the function f around g(x)".

We consider as composite functions not only those obtained when two different functions f(x) and g(x) are combined with each other, but also when a function f(x) is combined with itself, i.e. f ◦ f(x); when a function f(x) is multiplied or divided by a constant, i.e. c · f(x); when a function f(x) is raised to a certain power n, i.e. [f(x)]n, etc.

Composite functions have the following properties:

1. For any function f(x)
   f(c·x) ≠ c ∙ f(x)
   where c is a constant.
2. If f(x) and g(x) are functions, then in general
   f ∘ g(x) = g ∘ f(x) = x
   Thus, in general, the composition of functions is not commutative.
3. Composite functions are associative in the sense that if we have three functions f(x), g(x) and h(x), then
   f ∘ (g ∘ h) = (f ∘ g) ∘ h
4. If f(x) and g(x) are inverse functions in the sense that f(x) = g - 1(x), then
   f ∘ [g(x)] = g ∘ [f(x)] = xⁿ
   where n is an integer.
5. The composition of one-to-one (injective) functions is also one-to-one (injective).
6. The composition of surjective (onto) functions is also surjective (onto).
7. If f ◦ g(x) = g ◦ f(x), then f(x) = g(x).

Evaluating a composite function means finding the value of that composite function for a given input x.

It is possible to compose a function with itself. Suppose f(x) is a function, then the composition of function f with itself will be (f ∘ f) (x) = f [f(x)].

We can also find the value of a composite function by considering the two original functions one by one. Thus, if we want to find f ◦ g (x) for x = a, then we find first g(a) = b, then we calculate f(b) (a and b are numbers).

The composition of functions f(x) and g(x) is schematically shown below.

We can find the value of a composite function at a given point even when the individual functions f(x) and g(x) are not explicitly given but only shown on the graph. In this case, we can use the information provided in the graph as a shortcut for finding values of each function or the resulting composite functions without having the need to know the exact formulas of the given functions. When doing this, we must take into account the fact that if the ordered pair (x, y) is a point of the function f(x), then f(x) = y. Given this, we follow the procedure below to calculate f ◦ g(a) for x = a:

1. We find g(a) first so that the y-coordinate of g(x) for x = a is identified.
2. Then, we find f[g(a)], i.e. the y-value of f(x) that corresponds to g(a).

Sometimes the information about functions is given by tables. So we apply the same procedure as when dealing with graphs. The only difference is that now we have to identify the information from the table, not from the graph.

If we have two functions g : X → Y and f : Y → Z then f ∘ g : X → Z. This means the domain of f ∘ g is X and its range is Z.

The procedure described below is used to determine the domain of a composite function f ◦ g when the two individual functions are defined algebraically.

1. First, we find the domain of the inner function g(x) (say the set A).
2. Then, we find the domain of the composite function f ◦ g(x) (say the set B).
3. Last, we find the domain D of f ◦ g(x) obtained from the common elements of A and B (in set theory this new set is called intersection).

The range R of a composite function is identified in the same way as for simple functions, i.e. by checking for any restrictions that give not allowed values. However, in composite functions this procedure is carried out after making all operations, that is after obtaining the final form of the function.