The 42nd Period | STEM Education Hub

Use this as a quick reference for reading f(g(x)), domains, order, the identity function, building and decomposing compositions, and transformations.

🧭 Plot Summary

A composition feeds the output of one function into another. The composite function f∘g does g first and then f, so (f∘g)(x) = f(g(x)). You will evaluate compositions from tables, graphs, formulas, and words, notice where they are undefined, and see that order matters: f∘g and g∘f are typically different. You will meet the identity function I(x) = x, build new formulas by substitution, decompose a function into simpler parts, and see translations and dilations as compositions.

What you'll do in this lesson

Evaluate f(g(x)) from tables, graphs, formulas, and words, and notice when it is undefined.
Show that order matters, and meet the identity function that acts like 0 and 1.
Build a composition by substitution, and make a table or graph of it.
Relate two quantities that are not directly linked by a formula.
Decompose a function into simpler pieces and check that they rebuild the original.
See translations and dilations as compositions with x + k and kx.

Why it matters

Converting units, applying a discount and then a coupon, or turning time into radius into area all chain one rule into another. Composition is also the setup for the next topic: a function and its inverse compose to give the identity, which is the key to logarithms.

✅ Self-Check Before You Move On

Check off each item as you get there. These aren't grades — they're your own signal.

I can explain that (f∘g)(x) = f(g(x)) uses the output of g as the input of f, and I can find its domain.I can evaluate a composition from a table, a graph, a formula, or a verbal description.I can show with an example that f∘g and g∘f are typically different.I can explain what the identity function f(x) = x does when it is composed with another function.I can use composition to relate two quantities that are not directly related by a formula.I can build a formula for f(g(x)) by substituting g(x) for every x in f.I can build a table or graph of f∘g by calculating values of f(g(x)).I can rewrite a function as a composition of simpler functions, and check my answer.I can describe translations and dilations as compositions with g(x) = x + k and g(x) = kx.

🔬 Deep Dive

Your textbook for this topic: composite functions, evaluating, order, the identity, building by substitution, tables and graphs, decomposing, and transformations — with interactive tools.

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🧭 Guided Practice

Step through worked examples for every essential knowledge statement in 2.7.A, 2.7.B, and 2.7.C. Two or three examples each, revealed one step at a time.

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📝 Assignment

20 questions covering the whole lesson. Answer each one and get feedback right away, including the most likely reason behind a wrong choice.

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📌 AP Exam Note

Expect to evaluate a composition from a table, a graph, or a formula, to say where it is undefined, and to build a formula for f(g(x)) or g(f(x)). Compositions also appear in context: the output of one function in some units becomes the input of the next. Be ready to decompose a function and to explain why f∘g and g∘f differ.