Functions
A function is just a rule that takes an input and returns exactly one output, and on the GRE it usually shows up as f(x) notation or as an invented symbol you have never seen before. The exam is not testing whether you memorised graphs of exotic functions — it is testing whether you can follow a rule carefully. Nine times out of ten the "hard" function question is simple substitution done under time pressure, and the trap is a sign, an order-of-operations slip, or plugging into the wrong place. The one genuinely GRE-specific idea is the custom operator: the test defines a brand-new symbol like x ◊ y = x² − y and then asks you to compute with it. It looks intimidating and it is deliberately designed to — but it is only substitution. This chapter covers the notation, evaluating and composing functions, reading off domain and range, and those made-up-symbol questions, each with the fastest reliable method and the traps that cost points.
Topics
⚡ GRE shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- f(3) means "run the rule on 3" — the parentheses are not multiplication. Wrap every substitution in parentheses to kill sign errors.
- Compose inside-out: for f(g(x)), compute g first as a number, then apply f. Working numerically beats building the big composite formula.
- For domain, check just two things: no denominator equals 0, and nothing under a square root is negative. That covers almost every GRE case.
- A squared term or an absolute value is always ≥ 0, so its minimum is 0 — that instantly gives the range of things like x² + 5 or |x| − 2.
- A custom symbol is just a substitution rule. Read the definition, plug the numbers into the exact positions, and mind the order — x ◊ y ≠ y ◊ x in general.
- For nested custom operators, evaluate the innermost bracket to a single number before applying the operator again.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Reading f(x) as f times x and multiplying, instead of applying the rule.
- Composing in the wrong order — computing g(f(x)) when the question asks for f(g(x)).
- Sign slips when the input is negative: for f(x) = x² − 2x, f(−3) = 9 + 6 = 15, not 9 − 6.
- Forgetting to exclude values that make a denominator zero when stating the domain.
- Assuming a custom operator is symmetric — swapping the two inputs changes the result unless the rule is genuinely symmetric.
- Squaring or negating the wrong term in a custom rule (e.g. a + b² is not (a + b)²).
📈 GRE exam insight & question patterns
The GRE defines an unfamiliar symbol, say x ⊗ y = x² − y, then asks for a value. How should you treat it?
As pure substitution. The definition is the entire rule; e.g. 4 ⊗ 3 = 16 − 3 = 13. Its strangeness is only presentation.
Quantitative Comparison — f(x) = x². Quantity A: f(−4). Quantity B: f(3). Which is greater?
f(−4) = 16 and f(3) = 9, so Quantity A is greater. Squaring makes the negative input win — do not assume 3 > −4 decides it.
Numeric Entry — if f(x) = 2x − 5 and f(a) = 9, enter a.
Solve 2a − 5 = 9 → 2a = 14 → a = 7. Function questions often run backwards from an output to the input.
A composition question gives f and g as a table of values rather than formulas. How do you handle it?
Look up the inner value in the table first, then look up the outer function of that result — the mechanics are identical to formula-based composition.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara GRE 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 | 8 cards |
Formula Reference Sheet
Notation & evaluation
| Function value | f(a) = replace every x in the rule with a |
|---|---|
| Composition | f(g(x)) = evaluate g first, then feed into f |
| Custom operator | a ⊕ b = (whatever rule the problem defines) |
Domain & range
| Domain | all inputs x that are allowed |
|---|---|
| Range | all outputs f(x) that can result |
| Denominator rule | exclude any x making a denominator 0 |
| Even-root rule | the quantity under a √ must be ≥ 0 |