Functions • Topic 2 of 4

Types of Functions

CAT classifies functions along three axes. Symmetry: f is even if f(−x) = f(x) (symmetric about the y-axis, e.g. x², cos x, |x|) and odd if f(−x) = −f(x) (symmetric about the origin, e.g. x³, sin x, x). Most functions are neither, and a handy fact is that any function splits into an even part plus an odd part. Periodicity: f is periodic with period T if f(x + T) = f(x) for the smallest such T > 0 — the standard CAT example is the fractional-part function {x}, which has period 1. Mapping behaviour: f is one-one (injective) if different inputs always give different outputs, tested by f(a) = f(b) ⇒ a = b; a strictly increasing or strictly decreasing function is automatically one-one. f is onto (surjective) if its range fills the entire co-domain. A function that is both is a bijection, the only kind that can be inverted. Watch the subtle trap: x² is one-one on [0, ∞) but not on all of ℝ, so the stated domain decides the answer.

✅ Solved examples

1. Classify f(x) = x³ − x as even, odd or neither.
f(−x) = (−x)³ − (−x) = −x³ + x = −(x³ − x) = −f(x). So f is odd.
2. Is f(x) = x² + 3, with domain ℝ, one-one?
f(2) = 7 and f(−2) = 7 but 2 ≠ −2, so f is not one-one (it is even, two inputs share an output).
3. Find the period of f(x) = {x}, the fractional part of x.
{x + 1} = {x} for every real x, and no smaller positive value works, so the period is 1.
4. Is f: ℝ → ℝ, f(x) = 2x − 5, a bijection?
It is one-one (slope ≠ 0, strictly increasing) and onto (every real y = 2x − 5 has x = (y + 5)/2). So it is a bijection.

✏️ Practice — try these, take hints as needed

1. Classify f(x) = |x| + cos x.
Test f(−x).
|−x| = |x| and cos(−x) = cos x.
Both pieces are even.
Even
2. Is f(x) = x³ + x one-one on ℝ?
Check whether it is strictly increasing.
Derivative-free: x³ and x both increase with x.
A strictly increasing function is one-one.
Yes, one-one
3. Period of f(x) = sin(2x)?
Base period of sin x is 2π.
A factor k inside divides the period by k.
Period = 2π/2.
π
4. Is f: ℝ → ℝ, f(x) = x², onto?
Range is the set of outputs.
A square is never negative.
Co-domain ℝ includes negatives, range does not.
No, not onto
5. f(x) = x² is one-one on which of [0, ∞) or ℝ?
On ℝ, x and −x collide.
Restrict so no two inputs share an output.
Non-negative inputs are distinct.
[0, ∞)

📝 Topic test — 8 questions

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