How to Evaluate Composite Functions

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  • 0:05 Painting in Art Class
  • 0:58 Defining Composite Functions
  • 1:22 Writing Composite Functions
  • 2:54 Evaluating Composite Functions
  • 4:11 Lesson Summary
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Lesson Transcript
Instructor: Tyler Cantway

Tyler has tutored math at two universities and has a master's degree in engineering.

Sometimes a math function depends on the outcome of another function. This lesson breaks down this chain reaction, called a composite function, and how to evaluate them.

Painting in Art Class

As part of a school project, Ashley and Brad were painting posters. The posters were white. Ashley had red paint and yellow paint. Brad had blue paint. They each painted many signs. After a while, they were tired of painting by themselves, so they began painting as a team. Ashley handed a white poster to Brad, and he painted it blue. For the next poster, Ashley painted it red and quickly handed it to Brad. When Brad added blue paint to the red paint, the poster became purple. Ashley painted the next poster yellow and gave it to Brad. When Brad added his blue paint, the poster became green.

When each person was painting one color, they got the same answer. But when Ashley painted first and Brad painted second, the final poster might get a completely different color depending on the colors they used.

Defining Composite Functions

In math we have special formulas, called functions, that tell us an answer when we plug in a specific number. However, like the paintings, we can put functions together so that one function gives us a different answer depending on the answer of another. Composite functions use the output of one function as the input of another. This is like a function within a function.

Writing Composite Functions

Let's look at two normal functions: f(x) = x + 2 and g(x) = 3x. If we wanted to make a composite function, we would have to put one function inside the other. To make a composite function where we put g(x) inside the function f(x), we can write it f(g(x)).

Notice that instead of simply putting an x in the function, we substitute the entire g(x) function. When we do that, we have f(g(x)) = (3x) + 2. On the left, you'll see that the g function is inside the f function. On the right, you'll see that instead of the x we substituted 3x.

We can do this other ways too. We could substitute the f function inside the g function. When you put the f function inside the g function, you get g(f(x)) = 3(x+2).

You can even make a composite function of itself. We can make f(f(x)) = (x+2) +2 or g(g(x)) = 3(3x). In each case, we take the whole function and substitute it where we see x.

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