Defined operations, composition and recursion
Follow a supplied definition exactly, track the order of function inputs, and build terms from a recurrence.
What you will tackle
- Evaluate newly defined binary operations and two-input functions
- Construct and evaluate a composed function
- Generate recursive sequences
- Write a recurrence with an initial condition
Definitions control the operation
A new symbol has only the meaning supplied: if a ★ b = a² − 2b, use that rule rather than ordinary multiplication. A function f(x,y) takes an ordered pair of inputs; substituting one value for each variable is different from multiplying f by x and y.
Composition and recurrence
The composition (f ∘ g)(x) = f(g(x)) applies g first, then f. Reverse order can give a different rule. A recursive sequence uses earlier terms, such as a₁ = 3 and aₙ₊₁ = 2aₙ + 1. An initial value is essential: the recurrence alone does not determine a unique sequence.
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Let a ★ b = a² − 2b, h(x,y) = 2x − y², f(x) = 2x + 1 and g(x) = x². Find 3 ★ 4, h(3,−2), f(g(x)) and g(f(2)).
Show a starting hint
Write the inner function's output before evaluating the outer function.
Compare with the explained answer
- 3 ★ 4 = 9 − 8 = 1; h(3,−2) = 6 − 4 = 2.
- Apply g then f: f(g(x)) = 2x² + 1.
- f(2) = 5, so g(f(2)) = 25; this differs from f(g(2)) = 9.
Answer / response criteria: 1; 2; 2x² + 1; 25.
Independent practice
Independent 1
Let a₁ = 3 and aₙ₊₁ = 2aₙ + 1. Find the next three terms. Write recurrences for 7,11,15,… and 3,6,12,…, each with its initial value.
Compare with the explained answer
- Starting at 3 gives 7,15,31 by repeatedly doubling and adding one.
- The arithmetic sequence has b₁ = 7 and bₙ₊₁ = bₙ + 4.
- The geometric sequence has c₁ = 3 and cₙ₊₁ = 2cₙ.
Answer / response criteria: 7,15,31; recurrences as stated.
Application practice
Application 1
An account starts at $100. Each year its current balance increases by 10%, then a $5 fee is deducted. Define B₀ and Bₙ₊₁, calculate two years, and express one year as a composition. Explain why reversing the operations changes the result.
Compare with the explained answer
- B₀ = 100 and Bₙ₊₁ = 1.1Bₙ − 5, giving B₁ = 105 and B₂ = 110.50.
- With g(x) = 1.1x and f(x) = x − 5, one year is f(g(x)).
- Reversing gives g(f(x)) = 1.1x − 5.5, since the fee would be deducted before growth.
Answer / response criteria: $105 then $110.50; recurrence 1.1Bₙ − 5; growth then fee.
What can you explain now?
An account starts at $100. Each year its current balance increases by 10%, then a $5 fee is deducted. Define B₀ and Bₙ₊₁, calculate two years, and express one year as a composition. Explain why reversing the operations changes the result.
Compare with the explanation
$105 then $110.50; recurrence 1.1Bₙ − 5; growth then fee.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide