Math Lesson 16.4.8 - Domain of a Composite Function

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Welcome to our Math lesson on Domain of a Composite Function, this is the eighth lesson of our suite of math lessons covering the topic of Composite Functions, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Domain of a Composite Function

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).

Example 7

Two functions f(x) = √(x - 3) and g(x) = 5 - x (for x ≤ 7) are combined to give the two resulting composite functions f ◦ g (x) and g ◦ f (x). Find the domain of f ◦ g (x).

Solution 7

The inner function for the given composite function f ◦ g (x) is g(x) = 5 - x. Its domain Y consists of all numbers smaller than or equal to 7. In symbols, we denote this set as Y = (-∞, 7].

Now, we have to identify the domain Z of the outer function f(x) = √(x - 3). Since the expression inside the square root cannot be negative, it is clear that the domain Z of f(x) consists of all numbers greater than or equal to 3. In symbols, we write this set as Z = [3, + ∞). In this way, we are able to identify the common elements of the two individual domains above is D = [3, 7].

You have reached the end of Math lesson 16.4.8 Domain of a Composite Function. There are 9 lessons in this physics tutorial covering Composite Functions, you can access all the lessons from this tutorial below.

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