Follow the Inner Function before the Outer One

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Let f(x)=x² and g(x)=2x+1. Find (f∘g)(2). Candidate answers: (A) 24 (B) 25 (C) 26 (D) 50.
• Independent check inputs: Let f(x)=x² and g(x)=2x+1. Find (f∘g)(3). Candidate answers: (A) 48 (B) 49 (C) 50 (D) 98.

Learn the idea

Function operations must respect input order and domain. A composition feeds one output into another function, while an inverse reverses a one-to-one relationship and exchanges the original domain and range. In f(g(x)), g acts on the supplied input before f acts on its result. This nested process generally differs from g(f(x)) even when both functions are simple.

Here is the method on the supplied teaching case. Composition applies g first: g(2)=2(2)+1=5. Then apply f: (5)²=25. Reversing the order would give g(f(2)) instead.

Read the innermost operation first. Composition is not multiplication of function values; record the intermediate output so the outer function receives the computed value rather than the original input.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Let f(x)=x² and g(x)=2x+1. Find (f∘g)(9). Candidate answers: (A) 360 (B) 361 (C) 362 (D) 722. Show the mathematical reason, not only the selected answer. Expected result: 361. Composition applies g first: g(9)=2(9)+1=19. Then apply f: (19)²=361. Reversing the order would give g(f(9)) instead.

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