Cogito
AP Precalculus · Chapter 4 · Lesson 3
Composition and Decomposition
Building functions, and taking them apart.
12 problems · about 22 minutes · AP Precalculus 1.13
What this lesson teaches
The student composes functions, determines the domain of a composite, and decomposes a function.
- f(g(x)) means do g first, and the order matters.
- A composite domain needs x in g and g(x) in f.
- Decomposing a function names the layers the Chain Rule will later use.
Warm Up
Straightforward practice. Get the method working first.
5 problemsf(x) = 3x and g(x) = x + 2. What is f(g(4))?
Answer 18
Why 18.
Why find a composite domain before simplifying?
Answer Simplifying can hide a restriction the original composite had.
Why The restriction disappears from the simplified form.
f(x) = x + 1, g(x) = 3x. What is f(g(2))?
Answer 7
Why g(2) = 6, then add 1.
Same functions. What is g(f(2))?
Answer 9
Why f(2) = 3, then triple it.
f(x) = √x, g(x) = x − 9. Smallest x in the domain of f(g(x))?
Answer 9
Why x − 9 ≥ 0.
Build It Up
The same ideas with more to keep track of.
3 problemsf(x) = 1 ÷ x, g(x) = x − 2. Which x must be excluded from f(g(x))?
Answer 2
Why The inner output cannot be zero.
For √(3x + 1), what is the inner function at x = 5?
Answer 16
Why 3(5) + 1.
f(x) = x², g(x) = x + 1. What is f(g(3))?
Answer 16
Why g(3) = 4, then square.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the steps for finding a composite domain in order.
Answer 1. Find the domain of the inner function. 2. Require the inner output to lie in the outer domain. 3. Combine both restrictions. 4. Only then simplify the expression.
Why Simplifying last is what preserves the hidden restriction.
f(x) = 2x, g(x) = x². What is f(g(3))?
Answer 18
Why g(3) = 9, then double.
The Composition: f(x) = 2x and g(x) = x + 3. What is f(g(4))?
Answer 14
Why 14.
The Domain: f(x) = √x and g(x) = x − 5. What is the smallest x in the domain of f(g(x))?
Answer 5
Why 5.