Combining and Composing Functions

Building complicated models from simple parts.

12 problems · about 21 minutes · F-BF.A.1

What this lesson teaches

The student combines functions arithmetically and by composition, tracking the domain.

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.

Straightforward practice. Get the method working first.

5 problems

f(x) = 3x and g(x) = x + 2. What is f(g(4))?
Answer 18
Why 18.
What must hold for a composition to make sense in context?
Answer The inner output units must match the outer input units.
Why The units must connect.
f(x) = 2x and g(x) = x + 3. What is f(g(4))?
Answer 14
Why g(4) = 7, then double.
f(x) = x² and g(x) = x + 1. What is f(g(3))?
Answer 16
Why g(3) = 4, then square.
f(x) = 3 and g(x) = 5. What is (f + g)(x)?
Answer 8
Why 3 + 5.

The same ideas with more to keep track of.

3 problems

f(x) = √x and g(x) = x − 9. Smallest x in the domain of f(g(x))?
Answer 9
Why x − 9 ≥ 0.
A quotient f ÷ g. Which x values are excluded? 1 where g is zero, 2 where f is zero.
Answer 1
Why A zero denominator.
u(h) = 12h. How many units in 5 hours?
Answer 60
Why 12 × 5.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Put the steps for 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 preserves the hidden restriction.
f(x) = x + 5 and g(x) = 2x. What is f(g(3))?
Answer 11
Why g(3) = 6, then add 5.
The Composition: u(h) = 12h and c(u) = 5u. What is c(u(2))?
Answer 120
Why 120.
The Order: f(x) = 2x and g(x) = x + 3. What is g(f(4))?
Answer 11
Why 11.