Math Concept: Function Composition

By now, you should have seen my first post on functions. This section's concept is relatively simple and easy to understand. To put it simply, function composition is when one function is put into another function.

Function composition often takes its form like this: f(g(x)). In this form, it is basically saying that with an "x" input into the function of "g," the output for that function becomes the input for the function "f."

Function composition problems usually come with two different or even identical functions. Then it will tell you what the "x" value is, and you substitute that "x" value in for the function that is written out.

ex.
f(x) = 1/3x + 5
g(x) = 3x - 4

You are given two functions in this example, and the question might be something along the lines of: f(g(3))

Start out first with the g(3) part. Substitute 3 in for all x's in the function.
g(3) = 3(3) - 4
g(3) = 9 - 4
g(3) = 5

Now you have an output for the function g(x). This output becomes the input for the function of f(x).

f(5) = 1/3(5) + 5
f(5) = 5/3 + 5
f(5) = 1 2/3 + 5
f(5) = 6 2/3 or f(5) = 20/3

On occasion, you might come across a question asking for f(f(x)). This may seem confusing at first, but it follows the exact same rules as the scenario explained above.

ALTERNATIVE METHOD

An alternative method to solving function composition problems is simply plugging in the entire equation into the other.

ex.
f(x) = 10x - 3
g(x) = 3x - 4
find f(g(x))

A different approace to this problem is to simply plug the entire function into the other. f(g(x)) is stating that with the function g(x) as the "x" value for function f(x), f(g(x)) is the output.

f(g(x))
f(3x - 4) = 10(3x - 4) - 3
f(3x - 4) = 30x - 40 - 3
f(3x - 4) = 30x - 43

After this, you may plug in any value for "x" as the question requests and you will receive the same answer as the previous method.

Good luck! Let me know if you would like me to answer any questions about this concept or if I could clarify on something for you, drop a comment! :)