Algebra II · 35 minutes
Suggested grades 10–12 · Unit 3: Combining and reversing functions · Lesson 1 of 2
Composition and the Order of Operations
Track an intermediate output through two functions.
Mr. S Teaches · Original starter lesson · HS-ALG2-05 · Grade range is planning guidance.Watch, notice, discuss
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Introduction to function composition | Functions and their graphs | Algebra II | Khan Academy
By Khan Academy
Build the input-output meaning of applying one rule after another, then compare both orders in the lesson.
Open on YouTube: Introduction to function composition | Functions and their graphs | Algebra II | Khan Academy (opens a new tab) ↗Decomposing Functions - Composition of Functions
By The Organic Chemistry Tutor
An extension choice: unpack a composite rule into its inner and outer functions and discuss input restrictions.
Open on YouTube: Decomposing Functions - Composition of Functions (opens a new tab) ↗Why can reversing the order of the same two functions change both the output and allowed input?
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Learning objective
Compute and interpret two compositions and identify a relevant input restriction.
Before you begin
- Function notation
- Substitution
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- composition
- Using one function’s output as another function’s input.
- intermediate value
- The output passed between two stages.
- domain
- Inputs allowed by a function and its context.
Understand the concept
A fictional print order applies a $4 setup charge with f(x)=x+4 and a10% discount with g(x)=0.9x. The order matters: g(f(x)) discounts the setup charge, while f(g(x)) does not. Composition is not multiplication. A valid input must pass through every stage, including any square-root or denominator restrictions.
Teach it step by step
- Trace x=20 through f then g and write the intermediate amount.
- Trace the same x through g then f and explain the contextual difference.
- Derive both composite expressions by substitution rather than multiplying formulas.
- For h(x)=√x and k(x)=x−3, find the real domain of h(k(x)).
Make the thinking visible
Worked example or model response
g(f(20))=0.9(24)=21.60, whereas f(g(20))=18+4=22. The expressions are 0.9x+3.6 and 0.9x+4. For h(k(x))=√(x−3), x≥3.
Guided practice
For p(x)=2x−1 and q(x)=x², compute p(q(3)) and q(p(3)).
Independent practice
1. Write p(q(x)) and q(p(x)). 2. For r(x)=1/x and s(x)=x+2, state the excluded input of r(s(x)).
Quick assessment
Why does f(g(x)) generally differ from g(f(x))?
Answer key & assessment criteria
Answers and teaching guidance
- Guided: p(9)=17; q(5)=25.
- Independent 1:2x²−1 and(2x−1)². 2: x≠−2.
- Quick check: The functions apply different transformations in a different order; their intermediate inputs need not be equal.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Use input→stage 1→intermediate→stage 2→output boxes.
Extension challenge
Find two nonconstant functions that do commute and verify both compositions algebraically.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSF.BF.A.1.c
Common Core · High school · Functions
(+) Additional preparation for advanced study
Compose functions and interpret their stage order and allowed inputs.
Read the official standard ↗Additional preparation (+). This selected composition exercise is an instructional Algebra II placement, not a universal course requirement.
Verified 2026-11-04. Selected contribution, not full attainment.