Composite functions
A composite function is a function created 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).
For the functions f(x) and g(x), when g(x) is used as the input of f(x), the composite function is written as:
(f ∘ g)(x)
The ∘ symbol denotes a composite function - it looks similar to the multiplication symbol, ⋅, but does not mean the same thing. Composite functions can also be written in a different way: (f ∘ g)(x) is equivalent to f(g(x)). Similarly (g ∘ f)(x) and g(f(x)) are also equivalent.
Order of composite functions
It is important to note that (f ∘ g)(x) is not equivalent to (g ∘ f)(x;); the notation is not interchangeable and (f ∘ g)(x) will yield different results.
You can use composite functions to check if two functions are inverses of each other because they will follow the rule:
(f ∘ g)(x) = (g ∘ f)(x) = x
How to solve composite functions
To solve the composite of two functions, replace every x in the outer function with the equation for the inner function (the input).
Example
Given: f(x) = 4x^{2} + 3; g(x) = 2x + 1, find (f ∘ g)(x).
(f ∘ g)(x) | = | f(g(x)) = 4(2x + 1)^{2} + 3 |
= | 4(4x^{2} + 4x + 1) + 3 | |
= | 16x^{2} + 16x + 7 | |
(f ∘ g)(x) | = | 16x^{2} + 16x + 7 |
(g ∘ f)(x) | = | g(f(x)) = 2(4x^{2} + 3) + 1 |
= | 8x^{2} + 6 + 1 | |
(g ∘ f)(x) | = | 8x^{2} + 7 |
Just like with inverse functions, you need to apply domain restrictions as necessary to composite functions. The composite of two functions f(x) and g(x) must abide by the domain restrictions of f(x) and g(x). In this example, both functions had domains of all real numbers, so their composite functions did not have any domain restrictions either.
Let's look at an example where domain restrictions apply.
Example
Given: ; , find (f ∘ g)(x).
Although g(x) = x^{2} has a domain of all real numbers, has a domain of [0, ∞). Therefore, the composite function (f ∘ g)(x) and (g ∘ f)(x) both have a domain restriction of [0, ∞).
Domain of composite functions
The domain of a composite function is the intersection of the domains of the functions involved. In other words, to find the domain of a composite function (f ∘ g)(x),
- Find the domain of g(x).
- Find the domain of f(x).
- Find the x inputs in the domain of g that are also in the domain of f. These x inputs comprise the domain of (f ∘ g)(x).
The example above demonstrates this.
Properties of composite functions
Below are some properties of composite functions:
- The inverse of the composition of two functions is the composition of the inverse of both functions: (f ∘ g)^{-1} = g^{-1} ∘ f^{-1}
- The composition of 2 one-to-one functions is also one-to-one.
- Composite functions are associative: (f ∘ g) ∘ h = f ∘ (g ∘ h).
- Composite funtions are usually not commutative (f ∘ g)(x) ≠ (g ∘ f)(x).
- A composite function made up of at least one even function will be even.
- A composite function made up of all odd functions will be odd.