Combining and Composing Functions

Algebra Foundation

Algebra 2

Explore advanced functions, complex numbers, polynomials, rational and radical expressions, exponentials, logarithms, conics, sequences, introductory trigonometry, probability, and statistics.

Learn with original explanations and worked examples. OpenStax Intermediate Algebra 2e was consulted for topic coverage.

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Algebra 2 · Unit 1 · Topic 1.3

Combining and Composing Functions

Composition (f ∘ g)(x) = f(g(x)) uses the output of g as the input of f.

Definition and explanation

Composition (f ∘ g)(x) = f(g(x)) uses the output of g as the input of f. Its domain includes only inputs allowed by g whose outputs are also allowed by f.

Worked example

Let f(u) = √u and g(x) = x − 6. Then (f ∘ g)(x) = √(x − 6), with x ≥ 6. Reversing the order gives (g ∘ f)(x) = √x − 6, with x ≥ 0; these are different functions.

Common mistake

Function composition is not multiplication and its order usually matters.

Key takeaway

Composition (f ∘ g)(x) = f(g(x)) uses the output of g as the input of f.

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.