Functions
A function is just a rule that takes each input and hands back exactly one output. That single restriction — one input, one output — is what separates a function from an ordinary relation, and it quietly drives most of what CAT tests here. The exam rarely asks you to "draw the graph"; instead it hides functions inside questions about domain restrictions, symmetry, repeated composition, and inverses, then rewards the student who can reason about the rule without plotting a single point. Functions also tie the whole Algebra block together: quadratics are functions, modulus and greatest-integer expressions are functions, and many maxima-minima problems are really "find the range of f". This chapter builds that fluency in four moves. First, finding the domain and range — where the rule is allowed to live and what values it can return. Second, classifying functions as even, odd, periodic, one-one or onto. Third, composing functions, f(g(x)), and unwinding repeated composition. Fourth, inverting a function and reading f-inverse off a graph. Threaded through all of it are functional-equation problems — the f(x) + f(1/x) and f(x + y) = f(x) + f(y) style questions that look exotic but fall to a few standard substitutions. Master these and you stop fearing the symbol f and start using it.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Domain checklist: √ needs inside ≥ 0, denominator needs ≠ 0, log needs argument > 0 — then intersect all conditions.
- Range fast: complete the square for quadratics, or set y = f(x), solve for x, and demand x be real (discriminant ≥ 0).
- Even/odd in one line: f(−x) = f(x) is even, f(−x) = −f(x) is odd; replace x by −x and compare, do not plot.
- Repeated composition: compute f, f₂, f₃ … until it cycles, then reduce the iterate count modulo the period (e.g. f₁₀₀ with period 3 → f₁).
- Inverse recipe: y = f(x), swap x and y, solve for y. Linear ax + b inverts to (x − b)/a instantly.
- Spot involutions: 1/x, a − x and (ax + b)/(cx − a) satisfy f(f(x)) = x, so each is its own inverse — no algebra needed.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Treating f(g(x)) and g(f(x)) as equal — composition is not commutative, so always work inside out.
- Calling x² one-one on all of ℝ; it is only one-one once the domain is restricted to [0, ∞) (or (−∞, 0]).
- Confusing the reciprocal 1/f(x) with the inverse f⁻¹(x) — they are completely different objects.
- Forgetting strict inequalities: a log argument must be > 0, and a root in a denominator must be > 0, not just ≥ 0.
- Trying to invert a function that is not a bijection, or ignoring that the inverse swaps the domain and range.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Functions when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
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) |