Math Lesson 16.4.1 - The Meaning of Composite Function

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Welcome to our Math lesson on The Meaning of Composite Function, this is the first 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.

The Meaning of Composite Function

Let's get back to the examples above, where the task was to combine the two functions f(x) and g(x) in different ways. In the specific case, replacing the variable x of the function f(x) with the entire expression of g(x) means inserting 2x instead of x in the function f(x) = x^x. In this way, we obtain

On the other hand, replacing the variable x of the function g(x) with the entire expression of f(x) means inserting x^x instead of x in the function g(x) = 2x. In this way, we obtain

In both cases, we obtained two composite functions. However, as you see, the resulting composite functions are not the same. This means the combination procedure is important in such cases.

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.

Example 1

The functions

and

are given. Find the following:

1. 2[f(x)]²
2. -3/g(x)
3. [f(x)]² + 2g(x)
4. f ◦ g(x)
5. g ◦ f(x)

Solution 1

1. From the BEDMAS (PEMDAS) rule, it is known that powers come before multiplications in the order of operations. Therefore, we have to find [f(x)]² first and then multiply it by 2. Thus,
   [f(x)]² = (√(3x - 1))²
   = 3x - 1
   for x ≥ 1/3
   If the above condition is not highlighted, the function [f(x)]² will have no restrictions, which is wrong. However, let's focus on the composition of the function. Now, we have
   2[f(x)]² = 2 ∙ (3x - 1)
   = 6x - 2
   again, for x ≥ 1/3.
2. We have
   -3/ g(x)/ = -3/1/x + 2
   = -3/1 ∙ x + 2/1
   = -3x - 6
3. We have found the value of [f(x)]² at (a). Thus, for x ≥ 1/3, we have
   [f(x)]² + 2g(x)
   = 3x - 1 + 2 ∙ 1/x + 2
   = 3x - 1 + 2/x + 2
4. We have
   f ∘ g(x) = f[g(x)]
   = √3 ∙ 1/x + 2 - 1
   = √3/x + 2 - 1
5. We have
   g ∘ f(x) = g[f(x)]
   = 1/√3x - 1 + 2

You have reached the end of Math lesson 16.4.1 The Meaning of 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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