**GRE Algebra: Functions – Notation, Composition, Domain, Graphs**

GRE Algebra: Functions – Notation, Composition, Domain, Graphs

Complete Study Guide for High-Scoring Candidates

What This Is

Functions are a core GRE algebra topic, appearing in ~5–7 questions per test (Quantitative Reasoning). Mastery here boosts your score by 5–10 points because: - Notation & composition test your ability to manipulate abstract symbols—a key GRE skill.
- Domain & range questions exploit common misconceptions (e.g., square roots, denominators).
- Graph interpretation appears in Quantitative Comparison (QC) and Multiple-Choice formats, often as a time trap.

Real GRE-Style Example:
If ( f(x) = \sqrt{x + 3} ) and ( g(x) = 2x - 1 ), what is the domain of ( f(g(x)) )? (A) ( x \geq -1 ) (B) ( x \geq -3 ) (C) ( x \geq \frac{1}{2} ) (D) ( x \geq 1 ) (E) All real numbers

(Answer: A. Solution below.)

Key Concepts & Techniques

1. Function Notation (( f(x) ))
2. What it is: ( f(x) ) means "a rule that assigns an output to input ( x )."
3. When to use: Any time you see ( f(\text{something}) ), replace ( x ) with that "something."

4. GRE Trap: ( f(2x) ) is not the same as ( 2f(x) ). Example: If ( f(x) = x^2 ), then ( f(2x) = 4x^2 ), but ( 2f(x) = 2x^2 ).

5. Composition (( f(g(x)) ))

6. What it is: Plugging one function into another. Work inside-out: compute ( g(x) ) first, then plug into ( f ).
7. When to use: Questions like "If ( f(g(2)) = 5 ), what is ( g(2) )?" or domain problems (e.g., the example above).

8. GRE Trap: Order matters! ( f(g(x)) \neq g(f(x)) ) unless the functions are inverses.

9. Domain (Allowed Inputs)

10. What it is: All ( x ) values for which ( f(x) ) is defined.
11. When to use: Questions asking "For which values of ( x ) is ( f(x) ) defined?"
12. Key Rules:
    • Denominators: ( \frac{1}{x} ) → ( x \neq 0 ).
    • Square roots: ( \sqrt{x} ) → ( x \geq 0 ).
    • Logarithms: ( \log(x) ) → ( x > 0 ).

13. GRE Trap: Composite functions (( f(g(x)) )) require both ( g(x) ) to be in ( g )’s domain and ( g(x) ) to be in ( f )’s domain.

14. Range (Possible Outputs)

15. What it is: All possible ( y ) values ( f(x) ) can take.
16. When to use: Questions like "What is the range of ( f(x) = x^2 + 1 )?" (Answer: ( y \geq 1 )).

17. GRE Trap: Range is not always obvious. For ( f(x) = \frac{1}{x} ), the range is ( y \neq 0 ), but for ( f(x) = \sqrt{x} ), it’s ( y \geq 0 ).

18. Graph Interpretation

19. What it is: Reading ( f(x) ) values from a graph (e.g., "At ( x = 2 ), what is ( f(x) )?").
20. When to use: QC questions comparing ( f(a) ) vs. ( f(b) ), or "Which of the following could be ( f(x) )?" questions.
21. Key Skills:
    • Intercepts: Where the graph crosses the ( x )- or ( y )-axis.
    • Symmetry: Even functions (( f(-x) = f(x) )) are symmetric about the ( y )-axis; odd functions (( f(-x) = -f(x) )) are symmetric about the origin.
    • Asymptotes: Vertical (denominator = 0) or horizontal (end behavior).

22. GRE Trap: The GRE loves to test piecewise functions (e.g., ( f(x) = x ) for ( x \geq 0 ), ( f(x) = -x ) for ( x < 0 )). Always check the definition of the function.

23. Inverse Functions (( f^{-1}(x) ))

24. What it is: The function that "undoes" ( f ). If ( f(a) = b ), then ( f^{-1}(b) = a ).
25. When to use: Questions like "If ( f(3) = 7 ), what is ( f^{-1}(7) )?" (Answer: 3).

26. GRE Trap: Not all functions have inverses! Only one-to-one functions (pass the horizontal line test) do.

27. Linear vs. Nonlinear Functions

28. Linear: ( f(x) = mx + b ) (straight line, constant rate of change).
29. Nonlinear: Quadratic (( x^2 )), exponential (( 2^x )), etc. (curved, variable rate of change).
30. When to use: QC questions comparing ( f(x) ) at different points (e.g., "Is ( f(2) - f(1) ) greater than ( f(3) - f(2) )?").

Step-by-Step Strategy

For any function question, follow this process:

1. Identify the question type.
2. Is it about notation (e.g., ( f(2x) ))?
3. Composition (e.g., ( f(g(x)) ))?
4. Domain/range?

5. Graph interpretation?

6. Write down the given information.

7. For ( f(x) = \sqrt{x + 3} ), note the domain restriction (( x \geq -3 )).

8. For ( g(x) = 2x - 1 ), note it’s linear (no restrictions).

9. Apply the relevant rule.

10. Composition? Compute ( g(x) ) first, then plug into ( f ).
11. Domain? Check denominators, square roots, logs.

12. Graph? Locate ( x )-values and read ( y )-values.

13. Solve step-by-step.

14. For ( f(g(x)) ), first find ( g(x) = 2x - 1 ).
15. Then ( f(g(x)) = \sqrt{(2x - 1) + 3} = \sqrt{2x + 2} ).

16. Domain: ( 2x + 2 \geq 0 ) → ( x \geq -1 ).

17. Check for traps.

18. Did you mix up ( f(g(x)) ) and ( g(f(x)) )?
19. Did you forget to consider both functions’ domains in a composition?

20. For graphs, did you misread the ( x )- or ( y )-axis?

21. Eliminate wrong answers.

22. For domain questions, test edge cases (e.g., ( x = -1 ) in the example above).
23. For graph questions, sketch the function if unsure.

Fully Worked GRE-Style Example

Question:
If ( f(x) = \sqrt{x + 3} ) and ( g(x) = 2x - 1 ), what is the domain of ( f(g(x)) )?

Step 1: Identify the question type.
- Composition + domain.

Step 2: Write down the given information.
- ( f(x) = \sqrt{x + 3} ) → Domain: ( x \geq -3 ).
- ( g(x) = 2x - 1 ) → Domain: all real numbers.

Step 3: Apply the composition rule.
- ( f(g(x)) = f(2x - 1) = \sqrt{(2x - 1) + 3} = \sqrt{2x + 2} ).

Step 4: Find the domain of ( f(g(x)) ).
- The expression under the square root must be ( \geq 0 ): ( 2x + 2 \geq 0 ) → ( 2x \geq -2 ) → ( x \geq -1 ).

Step 5: Check for traps.
- Did we forget ( g(x) )’s domain? No, it’s all real numbers.
- Did we mix up ( f(g(x)) ) and ( g(f(x)) )? No, the question asks for ( f(g(x)) ).

Step 6: Match the answer.
- Domain: ( x \geq -1 ) → Answer: A.

Common Mistakes

1. Mistake: Forgetting to check the domain of both functions in a composition.
2. Why it happens: Students focus only on the outer function (( f )) and ignore the inner function (( g )).

3. Correct approach: For ( f(g(x)) ), ensure ( g(x) ) is in ( g )’s domain and ( g(x) ) is in ( f )’s domain.

4. Mistake: Misapplying function notation (e.g., ( f(2x) = 2f(x) )).

5. Why it happens: Students confuse input substitution with output multiplication.

6. Correct approach: ( f(2x) ) means "replace ( x ) with ( 2x ) in ( f )." ( 2f(x) ) means "multiply ( f(x) ) by 2."

7. Mistake: Ignoring piecewise functions in graph questions.

8. Why it happens: Students assume the graph is a single smooth curve.

9. Correct approach: Always check if the function is defined differently for different ( x )-values (e.g., ( f(x) = x ) for ( x \geq 0 ), ( f(x) = -x ) for ( x < 0 )).

10. Mistake: Confusing domain and range.

11. Why it happens: Students mix up "allowed inputs" (domain) with "possible outputs" (range).

12. Correct approach: Domain = ( x )-values; range = ( y )-values.

13. Mistake: Overlooking asymptotes in domain questions.

14. Why it happens: Students forget that denominators cannot be zero.
15. Correct approach: For ( f(x) = \frac{1}{x-2} ), domain is ( x \neq 2 ).

GRE Traps & Timing

1. Trap: Composition Order
2. The GRE will give you ( f(g(x)) ) and ( g(f(x)) ) as answer choices. Always compute inside-out.

3. Example: If ( f(x) = x^2 ) and ( g(x) = x + 1 ), then:

   
   
   • ( f(g(x)) = (x + 1)^2 )
   • ( g(f(x)) = x^2 + 1 ) These are not the same!

4. Trap: Hidden Domain Restrictions

5. The GRE loves to test square roots and denominators in disguise.

6. Example: ( f(x) = \frac{\sqrt{x}}{x-1} ) has domain ( x \geq 0 ) and ( x \neq 1 ).

7. Trap: Graph Misinterpretation

8. The GRE may show a graph with unlabeled axes or non-integer points. Always:

   
   
   • Check the scale (e.g., is each tick mark 1 unit or 0.5 units?).
   • Look for open/closed circles (piecewise functions).

9. Timing:

10. Easy function question (notation/domain): 45–60 seconds.
11. Medium (composition/graph): 60–90 seconds.
12. Hard (piecewise/inverse): 90–120 seconds.
13. Flag and return if stuck—don’t waste time!

Quick Practice

1. Question:
   If ( f(x) = 3x + 2 ) and ( g(x) = x^2 - 1 ), what is ( f(g(2)) )?
   (A) 5
   (B) 7
   (C) 11
   (D) 14
   (E) 17

Answer: C
Solution: ( g(2) = 2^2 - 1 = 3 ). Then ( f(3) = 3(3) + 2 = 11 ).

1. Question:
   The graph of ( f(x) ) is a line passing through (0, 2) and (3, 0). What is ( f(f(3)) )?
   (A) -2
   (B) 0
   (C) 2
   (D) 4
   (E) 6

Answer: C
Solution: ( f(3) = 0 ) (from the graph). Then ( f(f(3)) = f(0) = 2 ).

Last-Minute Cram Sheet

1. Function notation: ( f(a) ) = plug ( a ) into ( f ). Not the same as multiplication!
2. Composition: ( f(g(x)) ) = compute ( g(x) ) first, then plug into ( f ).
3. Domain traps:
4. ( \sqrt{x} ) → ( x \geq 0 ).
5. ( \frac{1}{x} ) → ( x \neq 0 ).
6. ( \log(x) ) → ( x > 0 ).
7. Composite domain: Check both functions’ domains.
8. Graphs: Even = symmetric over ( y )-axis; odd = symmetric over origin.
9. Inverse functions: ( f(a) = b ) → ( f^{-1}(b) = a ). Only one-to-one functions have inverses.
10. Linear vs. nonlinear: Linear = straight line; nonlinear = curved.
11. Piecewise functions: Check the definition for different ( x )-values.
12. Asymptotes: Vertical = denominator = 0; horizontal = end behavior.
13. GRE loves: Square roots, denominators, and composition order traps. Always double-check!