Functions can be added, subtracted, multiplied and divided. The result is defined wherever both parts are.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Division adds a restriction
A quotient also excludes anywhere the denominator function is zero.
Composition
f(g(x)) runs one function into another. Order matters, and the two orders usually differ.
The composite domain
x must be in the domain of g, and g(x) must land in the domain of f. Both conditions apply.
Why models compose
Cost depends on units produced, and units depend on hours worked. Composing gives cost directly from hours.
Units must match up
The output units of the inner function must be the input units of the outer one, or the composition is meaningless.
Functions can be added and multiplied
Profit is revenue minus cost, both functions of quantity. Combining functions arithmetically builds models from parts, and the domain of the result is the overlap of the domains.
Composition feeds one into the other
f(g(x)) applies g first, then f. The notation reads right to left, which is a persistent source of error, and writing out the substitution avoids it.
Composition restricts the domain
x must lie in the domain of g and g(x) in the domain of f. The result is often narrower than either function alone, and that restriction has to be reported.
Where composition appears
Cost as a function of quantity, quantity as a function of time, so cost as a function of time. Chaining relationships is how multi-stage models are built, in economics and in science alike.
Step 2: Try It Yourself
Tap and try it out.
- Point(4, 2)
Step 3: Watch an Example
One step at a time.
Watch Ines Compose a Cost Model
A worker makes 12 units an hour, and each unit costs $5 in materials.
- Step 1
Units as a function of hours is u(h) = 12h.
Step 4: Your Turn
Practice makes it stick.
The Composition
Problem 1 of 2
u(h) = 12h and c(u) = 5u. What is c(u(2))?
The Order
Problem 2 of 2
f(x) = 2x and g(x) = x + 3. What is g(f(4))?
Build From Parts
1 of 8
f(x) = 2x and g(x) = x + 3. What is f(g(4))?
2 of 8
f(x) = x² and g(x) = x + 1. What is f(g(3))?
3 of 8
f(x) = 3 and g(x) = 5. What is (f + g)(x)?
4 of 8
f(x) = √x and g(x) = x − 9. Smallest x in the domain of f(g(x))?
5 of 8
A quotient f ÷ g. Which x values are excluded? 1 where g is zero, 2 where f is zero.
6 of 8
u(h) = 12h. How many units in 5 hours?
7 of 8
Put the steps for 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) = x + 5 and g(x) = 2x. 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
What must hold for a composition to make sense in context?
What You Learned
- Functions combine arithmetically wherever both are defined.
- A composition needs x in the inner domain and the inner output in the outer domain.
- In context, the inner output units must match the outer input units.