How to Solve: Functions Questions (GRE/GMAT) – Complete Guide

How to Solve: Functions Questions (GRE/GMAT) – Complete Guide

Score Impact: Functions questions appear 4-6 times per GRE or GMAT exam—mastering them can boost your Quant score by 5+ points and push you into the 90th+ percentile.

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The exam isn’t testing your ability to do math—it’s testing your ability to: 1. Decode function notation (e.g., f(x), g(a,b)) under time pressure. 2. Avoid misapplying rules (e.g., assuming f(a+b) = f(a) + f(b) when it’s not given). 3. Spot hidden constraints (e.g., domain restrictions, piecewise definitions).

ANATOMY OF THE QUESTION

Structure Breakdown

Representative Example (GRE/GMAT Style)

For all real numbers x, let f(x) = x² – 4x + 3. If f(a) = 0, which of the following could be the value of a? (A) -3 (B) -1 (C) 1 (D) 2 (E) 4

THE DECISION FRAMEWORK (Step-by-Step)

Run this process every time—no exceptions.

1. Read the stem first.
2. Underline the function definition (e.g., f(x) = x² – 4x + 3).

3. Circle any constraints (e.g., "for all real numbers x").

4. Identify the question type.

5. Evaluation: Find f(5).
6. Inverse: Find x where f(x) = 0.
7. Composition: Find f(g(x)).

8. Comparison: Is f(a) > f(b)?

9. Plug in or solve.

10. If given a value (e.g., f(5)), substitute directly.

11. If given an output (e.g., f(a) = 0), set up an equation and solve.

12. Check for traps.

13. Does the function have multiple solutions (e.g., quadratics)?

14. Are there domain restrictions (e.g., f(x) = √x implies x ≥ 0)?

15. Eliminate wrong answers.

16. Cross out options that violate constraints (e.g., negative inputs for √x).

17. For "could be" questions, test each option.

18. Confirm the answer.

19. If time permits, re-substitute to verify.

Worked Examples

Example 1 – Straightforward (Evaluation)

Let f(x) = 3x – 7. What is f(4)? (A) -19 (B) -5 (C) 5 (D) 12 (E) 19

Step-by-Step: 1. Stem: f(x) = 3x – 7 (linear function, no traps). 2. Question: Evaluate f(4). 3. Substitute: f(4) = 3(4) – 7 = 12 – 7 = 5. 4. Match: Option (C).

Time: <30 seconds.

Example 2 – Common Trap (Inverse with Multiple Solutions)

For all real numbers x, let f(x) = x² – 5x + 6. If f(a) = 0, which of the following could be a? (A) -3 (B) -2 (C) 1 (D) 2 (E) 5

Step-by-Step: 1. Stem: f(x) = x² – 5x + 6 (quadratic, so two solutions). 2. Question: Find a where f(a) = 0. 3. Set up equation: a² – 5a + 6 = 0. 4. Factor: (a – 2)(a – 3) = 0 → a = 2 or a = 3. 5. Check options: Only a = 2 is listed (Option D). 6. Trap: Option (E) is 5, which is f(5) = 25 – 25 + 6 = 6 (not zero).

Time: 45 seconds.

Example 3 – Hard Variant (Piecewise Function)

Let f(x) be defined as: f(x) = x + 2 if x ≤ 0 f(x) = x² if x > 0 What is f(f(-1))? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

Step-by-Step: 1. Stem: Piecewise function with two cases. 2. Question: Find f(f(-1)). 3. First, find f(-1):
- -1 ≤ 0 → use f(x) = x + 2.
- f(-1) = -1 + 2 = 1. 4. Now find f(1):
- 1 > 0 → use f(x) = x².
- f(1) = 1² = 1. 5. Final answer: 1 → Option (B). 6. Trap: Option (D) is 3, which is f(1) + 2 (misapplying the function).

Time: 60 seconds.

WRONG ANSWER PATTERNS

Common Mistakes

TIME STRATEGY

• Target time: 45–75 seconds per question.
• Skip if:
• The function is unfamiliar (e.g., f(x) = |x – 3| + 2).
• You’re stuck after 90 seconds.
• Minimum work:
• Substitute first, then eliminate.
• For "could be" questions, test 2-3 options before solving fully.

BACKSOLVING AND SHORTCUTS

1. Plug in numbers for "could be" questions.
2. Example: f(a) = 0 → test a = 1, 2, 3 instead of solving.
3. Eliminate impossible options first.
4. Example: f(x) = √x → eliminate negative inputs.
5. For composition (f(g(x))), work inside-out.
6. Find g(x) first, then f(g(x)).
7. Use symmetry for even/odd functions.
8. f(-x) = f(x) → even (e.g., x²).
9. f(-x) = -f(x) → odd (e.g., x³).

1-Minute Recap

"Functions questions test one thing: can you follow the rules? Not the math—the rules of the function. Here’s how to crush them every time:

1. Read the stem first. Underline the function—is it linear? Quadratic? Piecewise?
2. Plug in or solve. If they give you f(5), substitute. If they give f(a) = 0, set up an equation.
3. Check for traps. Multiple solutions? Domain restrictions? Piecewise cases?
4. Eliminate wrong answers. Cross out options that break the rules.
5. Confirm. Re-substitute if you have time.

Most mistakes happen when you assume—don’t. Stick to the definition, and you’ll get it right. Now go practice—timed!

Final Notes

• GRE vs. GMAT: GMAT functions are slightly harder (more piecewise, absolute value).
• Practice: Do 10-15 functions questions under timed conditions.
• Key Takeaway: Substitute first, eliminate second, confirm last.

Next Steps: 1. Drill evaluation (f(5)) and inverse (f(a) = 0) questions. 2. Master piecewise functions—they’re the hardest variant. 3. Review wrong answer patterns to spot traps faster.