The document discusses function composition, which is combining two or more functions into a new function by plugging the output of one function into the input of another. Specifically, it provides an example of composing the functions f(x) = 3x + 2 and g(x) = x - 1, finding the value of f(g(3)) by first applying g(3) to get 2, then applying f(2) to get the final answer of 8. It also explains that the second function listed is always plugged into the x variable of the first function to perform the composition.

1. Function Composition

2. Function Composition
What does composition mean?
◦ Combining two or more things to create something new.
What does function composition mean?
◦ Combining two or more functions, using a specific rule, to create a new function.

3. Function Compositions
We normally see function compositions either written as:
𝑓 𝑔 𝑥 or 𝑓°𝑔(𝑥)
Both are talking about the SAME THING
BUT, what do they MEAN?

4. Function Composition: Meaning
In general, for function compositions, we are simply plugging one function into
the other function.
Sounds confusing, but it is simple!

5. Function Composition: Meaning
In general, for function compositions, we are simply plugging one function into
the other function.
Let’s start simple:
𝑓 𝑥 = 3𝑥 + 2 and 𝑔 𝑥 = 𝑥 − 1
Find: 𝑓(𝑔 3 )

6. Function Composition: Meaning
𝑔 𝑥 = 𝑥 − 1 and 𝑓 𝑥 = 3𝑥 + 2
Find: 𝑓(𝑔 3 )
The number is plugged into the function it is listed closest to. The “g” function in the
above example. And we solve to get an answer.
𝑔 3 = 3 − 1 = 2
That answer is then plugged into the function farther away. The “f” function in the
above example.
𝑓 2 = 3 2 + 2 = 8

7. Function Composition
𝑓 𝑔 𝑥 or 𝑓°𝑔(𝑥)
The second function listed (the g function in the above examples) is ALWAYS
plugged into the x variable in the first function (the f function in the above
examples)

8. Example 1
Example:
𝑓 𝑥 = 𝑥2
𝑔 𝑥 = 2𝑥 − 1
𝑓°𝑔(𝑥)
This says: plug the g function into the f function.

9. Example 1
Example:
𝑓 𝑥 = 𝑥2
𝑔 𝑥 = 2𝑥 − 1
Step 1: rewrite 𝑓 𝑥 = 𝑥2 but change the x into parenthesis.
𝑓 𝑥 = ( )2
Step 2: put the g(x) function into the parenthesis.
𝑓 𝑥 = (2𝑥 − 1)2
Step 3: Simplify if possible.
𝑓°𝑔(𝑥)