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Composition of Functions: Definition & Examples

Instructor: DaQuita Hester

DaQuita has taught high school mathematics for six years and has a master's degree in secondary mathematics education.

What is a composition of functions and exactly how do you solve them? Find out and practice solving them in this lesson, then test your skills and knowledge with a quiz.

What is a Composition of Functions?

In mathematics, a function is like a machine. It performs a set of operations on an input in order to produce an output. Therefore, a composition of functions occurs when the output, or result, of one function becomes the input of another function.

For functions represented by f(x) or g(x), the composition would be represented by f(g(x)) or g(f(x)). It is important to know that f(g(x)) does not usually have the same value as g(f(x)), so order matters when calculating their composition. A general rule to keep in mind is to work from the inside out. The innermost function is the one that you should start with.

Practice

Let's begin our practice by looking at a few basic examples.

First, let f(x) = 2x and g(x) = x + 4. We are going to calculate the composition of f(g(3)). Since g(3) is written on the inside, we will start with that function first and substitute 3 into the equation for g(x). When we do, we see that g(3) = 3 + 4 = 7. Now, since g(3) = 7, we can see that f(g(3)) = f(7). From here, we are going to substitute 7 into the equation for f(x). We will see that f(7) = 2(7) = 14. So in conclusion, f(g(3)) = 14.

Let's try another one. For our second example, let f(x) = 2x + 1, and let g(x) = x^2. We are going to calculate g(f(-2)). Since f(-2) is written on the inside, we will start with that function. By substituting -2 into the equation for f(x), we see that f(-2) = 2(-2) + 1 = -4 + 1 = -3. So now, g(f(-2)) becomes g(-3), and we are ready to substitute -3 into the equation for g(x). We are able to see that g(-3) = (-3)^2 = 9. Therefore, g(f(-2)) = 9.

For our last example, we will use three functions. Let f(x) = x/2, g(x) = 2x - 5, and h(x) = 3x. We are going to calculate the composition of h(g(f(6))). The innermost function is f(x), so we will start there. We see that f(6) = 6/2 = 3. Since f(6) = 3, h(g(f(6))) becomes h(g(3)). From here, we will move to g(3) because it is now the innermost function. By substitution, g(3) = 2(3) - 5 = 6 - 5 = 1. Now that we know g(3) = 1, h(g(3)) becomes h(1), and we are ready for our last calculation. Through substitution, we see that h(1) = 3(1) = 3. This shows us that the overall composition of h(g(f(6))) = 3.

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