A composition feeds the output of one function into another. f(g(x)) means do g first, then f.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Order matters
f(g(x)) and g(f(x)) are usually different functions. Socks then shoes is not shoes then socks.
The domain of a composite
x must be in the domain of g, and g(x) must be in the domain of f. Both restrictions apply.
A hidden restriction
Simplifying can hide a restriction that the original composite had. Find the domain before simplifying.
Decomposition
Given √(3x + 1), the inner function is 3x + 1 and the outer is the square root. Naming the layers is what calculus later needs.
Why it matters here
The Chain Rule differentiates a composite layer by layer. Recognising the layers now is the preparation for it.
Building and taking apart
Composition builds complicated functions from simple ones. Decomposition runs it backwards, identifying √(x² + 1) as the square root applied to x² + 1. Both directions are examined.
Order and notation
f(g(x)) applies g first. The notation reads right to left, which is a persistent source of error. Writing out the substitution rather than doing it mentally is the reliable habit.
The domain shrinks
x must be in the domain of g, and g(x) must land in the domain of f. Composition usually restricts the domain further than either function alone, and that restriction must be reported.
Why decomposition matters
The chain rule differentiates a composition by working layer by layer, so recognising the layers is a prerequisite. Practising decomposition here is direct preparation for the hardest rule in differentiation.
Step 2: Try It Yourself
Tap and try it out.
- Point(4, 2)
Step 3: Watch an Example
One step at a time.
Watch Sana Find a Composite Domain
Sana has f(x) = √x and g(x) = x − 5, and needs the domain of f(g(x)).
- Step 1
The composite is √(x − 5).
Step 4: Your Turn
Practice makes it stick.
The Composition
Problem 1 of 2
f(x) = 2x and g(x) = x + 3. What is f(g(4))?
The Domain
Problem 2 of 2
f(x) = √x and g(x) = x − 5. What is the smallest x in the domain of f(g(x))?
Build and Break Apart
1 of 8
f(x) = x + 1, g(x) = 3x. What is f(g(2))?
2 of 8
Same functions. What is g(f(2))?
3 of 8
f(x) = √x, g(x) = x − 9. Smallest x in the domain of f(g(x))?
4 of 8
f(x) = 1 ÷ x, g(x) = x − 2. Which x must be excluded from f(g(x))?
5 of 8
For √(3x + 1), what is the inner function at x = 5?
6 of 8
f(x) = x², g(x) = x + 1. What is f(g(3))?
7 of 8
Put the steps for finding a composite domain in order.
- 1Require the inner output to lie in the outer domain.
- 2Combine both restrictions.
- 3Only then simplify the expression.
- 4Find the domain of the inner function.
8 of 8
f(x) = 2x, g(x) = x². What is f(g(3))?
Step 5: Quick Check
Show what you know.
Question 1 of 2
f(x) = 3x and g(x) = x + 2. What is f(g(4))?
Question 2 of 2
Why find a composite domain before simplifying?
What You Learned
- 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.