Composite Functions
Composition feeds the output of one function into another: (f∘g)(x) = f(g(x)), meaning apply g first, then f. The order matters — f(g(x)) is almost never the same as g(f(x)), so read the brackets carefully from the inside out. CAT loves two angles. First, plain evaluation: given f and g, compute f(g(x)) or a numeric value like f(g(2)) by substituting step by step. Second, repeated composition: define f₂(x) = f(f(x)), f₃ = f(f₂), and so on, then ask for f₁₀₀(x). The trick there is to compute the first few iterates and look for a cycle — many CAT functions (such as f(x) = 1/(1 − x) or f(x) = (x − 1)/(x + 1)) repeat with a short period, so f applied n times depends only on n modulo that period. A function that satisfies f(f(x)) = x is its own inverse, the period-2 case. Always track the domain too: g(x) must land inside the domain of f for the composite to be defined.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Definitions and tests
| Function rule | each x in domain → exactly one f(x) |
|---|---|
| Even function | f(−x) = f(x) (graph symmetric about y-axis) |
| Odd function | f(−x) = −f(x) (graph symmetric about origin) |
| Periodic function | f(x + T) = f(x) for least T > 0 |
| Composite function | (f∘g)(x) = f(g(x)) |
| Inverse condition | f(f⁻¹(x)) = x and f⁻¹(f(x)) = x |
CAT power-tools
| Domain of √(g(x)) | need g(x) ≥ 0 |
|---|---|
| Domain of 1/g(x) | need g(x) ≠ 0 |
| One-one (injective) test | f(a) = f(b) ⇒ a = b |
| Onto (surjective) test | range = co-domain |
| Inverse of linear f(x)=ax+b | f⁻¹(x) = (x − b)/a |
| Self-inverse / involution | f(f(x)) = x (e.g. f(x)=1/x, a−x) |