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GMAT Question Patterns: Functions and Custom Characters
Table of Contents:
• Regular and Nested Functions
• Custom Symbol Questions
• Function Domain and Range Questions
• Number Properties in Functions
• Theory
• Questions
Pattern #1: Regular and Nested Functions
- Involves basic function rules or plugging one function into another (like f(x), f(f(x)), g(f(2))).
Example 1.1 (Difficulty: sub-500 Level)
The function f is defined for all positive numbers x by \(f(x) = \frac{1}{x}\). What is the value of \(f(f(2) + f(\frac{1}{3})) \)?
A. 2/7
B. 2/3
C. 3/2
D. 7/3
E. 7/2
Question Discussion.
Example 1.2 (Difficulty: 505-555 Level)
The function g is defined for all nonzero numbers x by the equation \(g(x) = \frac{1}{2x}\). If x ≠ 0, what is g(g(x)) in terms of x ?
A. \(x\)
B. \(4x\)
C. \(\frac{1}{x}\)
D. \(\frac{1}{2x}\)
E. \(\frac{1}{4x^2}\)
Question Discussion.
Example 1.3 (Difficulty: 555-605 Level)
If f is a function defined for all k by \(f(k) = \frac{k^5}{16}\), what is f(2k) in terms of f(k)?
A. \(\frac{1}{8} f(k)\)
B. \(\frac{5}{8} f(k)\)
C. \(2 f(k)\)
D. \(10 f(k)\)
E. \(32 f(k)\)
Question Discussion.
Pattern #2: Custom Symbol Questions
- Uses special characters or made-up operations defined in the problem (like a □ b or ▲b). Solving means applying the rule exactly as given.
Example 2.1 (Difficulty: sub-500 Level)
The operation □ is defined for all a and b by a□b = a + b + ab. If a□b = 0 and b ≠ -1, what is a in terms of b?
A. \(\frac{b + 1}{b}\)
B. \(\frac{b - 1}{b + 1}\)
C. \(\frac{1}{b + 1}\)
D. \(\frac{b}{b + 1}\)
E. \(-\frac{b}{b + 1}\)
Question Discussion.
Example 2.2 (Difficulty: 505-555 Level)
For each positive integer n, let \(n^* = (2)(4)(6)...(2n)\) and \(n! = (1)(2)(3)...(n)\). Which of the following is equivalent to \(n^*÷n!\)?
A. (n - 1)!
B. n!
C. (n + 1)!
D. 2^n
E. 2^(n + 1)
Question Discussion.
Example 2.3 (Difficulty: 505-555 Level)
If x and y are positive real numbers, the operation △ is defined by \(x△y = \frac{xy}{x + y}\). Let a, b, and c, be positive real numbers. Which of the following is equivalent to \(\frac{1}{a}△(\frac{1}{b}△\frac{1}{c})\) ?
A. \(\frac{a + b + c}{abc}\)
B. \(\frac{ab + bc + ac}{abc}\)
C. \(\frac{1}{a + b + c}\)
D. \(\frac{ab + bc + ac}{a + b + c}\)
E. \(\frac{abc}{a + b + c}\)
Question Discussion.
Example 2.4 (Difficulty: 505-555 Level)
For all positive integers j and k, the operation k ◇ j is defined to be the product of j consecutive integers, beginning with k. For example, 6 ◇ 4 = 6 x 7 x 8 x 9. If a = 20 ◇ 20 and b = 21 ◇ 20, then \(\frac{a}{b}\) =
(A) \(\frac{21}{40}\)
(B) \(\frac{20}{39}\)
(C) \(\frac{20}{21}\)
(D) \(\frac{1}{5}\)
(E) \(\frac{1}{2}\)
Question Discussion.
Example 2.5 (Difficulty: 505-555 Level)
If ▲ is defined for all nonnegative integers b by \(▲b = (b + 1)^b\), what is the value of ▲(▲(▲0)) ?
A. 0
B. 2
C. 3
D. 4
E. 9
Question Discussion.
Example 2.6 (Difficulty: 505-555 Level)
For all positive integers x and y, the expression xΘy is defined as the least multiple of y that is greater than or equal to x. For example, 2Θ3 = 3 and 3Θ2 = 4. For how many different positive integers k is 20Θk = 30?
A. One
B. Two
C. Three
D. Four
E. Five
Question Discussion.
Pattern #3: Function Domain and Range Questions
- Focuses on where the function is defined (domain) and what values it can take (range).
Example 3.1 (Difficulty: 555-605 Level)
If f(x) = g(x - 1), where g(x) = |x| + 1, which of the following must be true?
(A) f(x) < 0
(B) f(x) = 0
(C) f(x) > 0
(D) f(x) = 2
(E) f(x) = g(x)
Question Discussion.
Example 3.2 (Difficulty: 555-605 Level)
The domain of the function \(f(x) = \frac{\sqrt{x - 1} }{x+1}\) is the set of all real numbers that are
A. greater than 1
B. greater than or equal to 1
C. not equal to -1
D. less than or equal to 1
E. less than 1
Question Discussion.
Example 3.3 (Difficulty: 655-705 Level)
The domain of the function \(f(x) = \sqrt{\sqrt{x+2} - \sqrt{4-x} }\) is the set of real numbers x such that
A. -2 ≤ x ≤ 1
B. -2 ≤ x ≤ 4
C. 1 ≤ x ≤ 2
D. 1 ≤ x ≤ 4
E. 2 ≤ x ≤ 4
Question Discussion.
Example 3.4 (Difficulty: 705-805 Level)
The range of a function is the set of all possible values of the function. If the function f is defined by \(f(x) = \frac{1}{x^2 + 1}\) for all real numbers x, the range of f is the set of
A. all real numbers
B. all real numbers except -1 and 1
C. all positive real numbers
D. all real numbers greater than or equal to 0 and less than or equal to 1
E. all positive real numbers less than or equal to 1
Question Discussion.
Pattern #4: Number Properties in Functions
- Involves rounding, greatest integer, or other number property logic applied within a function.
Example 4.1 (Difficulty: 505-555 Level)
For all numbers n, the symbol [n] denotes the greatest integer less than or equal to n. What is the value of [√2] + [√5] + [√7] ?
A. 3
B. 4
C. 5
D. 6
E. 8
Question Discussion.
Example 4.2 (Difficulty: 555-605 Level)
For all real numbers x, let \(\overline{x}\) denote the number obtained by rounding x to the nearest hundredth. If x = 21.9445, what is the value of \(10x−10\overline{x}\) ?
A. 0.005
B. 0.045
C, 0.095
D. 0.45
E. 0.95
Question Discussion.
Theory:
• How to Interpret Unfamiliar Symbols
• Functions on GMAT
• A Closer Look at GMAT Function Questions
Questions:
• Operations/functions defining algebraic/arithmetic expressions
• Rounding Functions Problems
• Symbols Representing Arithmetic Operation
• Various Functions Problems
• Hard Questions
• Medium Questions
• Easy Questions
Table of Contents:
• Regular and Nested Functions
• Custom Symbol Questions
• Function Domain and Range Questions
• Number Properties in Functions
• Theory
• Questions
Pattern #1: Regular and Nested Functions
- Involves basic function rules or plugging one function into another (like f(x), f(f(x)), g(f(2))).
Example 1.1 (Difficulty: sub-500 Level)
The function f is defined for all positive numbers x by \(f(x) = \frac{1}{x}\). What is the value of \(f(f(2) + f(\frac{1}{3})) \)?
A. 2/7
B. 2/3
C. 3/2
D. 7/3
E. 7/2
Question Discussion.
Example 1.2 (Difficulty: 505-555 Level)
The function g is defined for all nonzero numbers x by the equation \(g(x) = \frac{1}{2x}\). If x ≠ 0, what is g(g(x)) in terms of x ?
A. \(x\)
B. \(4x\)
C. \(\frac{1}{x}\)
D. \(\frac{1}{2x}\)
E. \(\frac{1}{4x^2}\)
Question Discussion.
Example 1.3 (Difficulty: 555-605 Level)
If f is a function defined for all k by \(f(k) = \frac{k^5}{16}\), what is f(2k) in terms of f(k)?
A. \(\frac{1}{8} f(k)\)
B. \(\frac{5}{8} f(k)\)
C. \(2 f(k)\)
D. \(10 f(k)\)
E. \(32 f(k)\)
Question Discussion.
Pattern #2: Custom Symbol Questions
- Uses special characters or made-up operations defined in the problem (like a □ b or ▲b). Solving means applying the rule exactly as given.
Example 2.1 (Difficulty: sub-500 Level)
The operation □ is defined for all a and b by a□b = a + b + ab. If a□b = 0 and b ≠ -1, what is a in terms of b?
A. \(\frac{b + 1}{b}\)
B. \(\frac{b - 1}{b + 1}\)
C. \(\frac{1}{b + 1}\)
D. \(\frac{b}{b + 1}\)
E. \(-\frac{b}{b + 1}\)
Question Discussion.
Example 2.2 (Difficulty: 505-555 Level)
For each positive integer n, let \(n^* = (2)(4)(6)...(2n)\) and \(n! = (1)(2)(3)...(n)\). Which of the following is equivalent to \(n^*÷n!\)?
A. (n - 1)!
B. n!
C. (n + 1)!
D. 2^n
E. 2^(n + 1)
Question Discussion.
Example 2.3 (Difficulty: 505-555 Level)
If x and y are positive real numbers, the operation △ is defined by \(x△y = \frac{xy}{x + y}\). Let a, b, and c, be positive real numbers. Which of the following is equivalent to \(\frac{1}{a}△(\frac{1}{b}△\frac{1}{c})\) ?
A. \(\frac{a + b + c}{abc}\)
B. \(\frac{ab + bc + ac}{abc}\)
C. \(\frac{1}{a + b + c}\)
D. \(\frac{ab + bc + ac}{a + b + c}\)
E. \(\frac{abc}{a + b + c}\)
Question Discussion.
Example 2.4 (Difficulty: 505-555 Level)
For all positive integers j and k, the operation k ◇ j is defined to be the product of j consecutive integers, beginning with k. For example, 6 ◇ 4 = 6 x 7 x 8 x 9. If a = 20 ◇ 20 and b = 21 ◇ 20, then \(\frac{a}{b}\) =
(A) \(\frac{21}{40}\)
(B) \(\frac{20}{39}\)
(C) \(\frac{20}{21}\)
(D) \(\frac{1}{5}\)
(E) \(\frac{1}{2}\)
Question Discussion.
Example 2.5 (Difficulty: 505-555 Level)
If ▲ is defined for all nonnegative integers b by \(▲b = (b + 1)^b\), what is the value of ▲(▲(▲0)) ?
A. 0
B. 2
C. 3
D. 4
E. 9
Question Discussion.
Example 2.6 (Difficulty: 505-555 Level)
For all positive integers x and y, the expression xΘy is defined as the least multiple of y that is greater than or equal to x. For example, 2Θ3 = 3 and 3Θ2 = 4. For how many different positive integers k is 20Θk = 30?
A. One
B. Two
C. Three
D. Four
E. Five
Question Discussion.
Pattern #3: Function Domain and Range Questions
- Focuses on where the function is defined (domain) and what values it can take (range).
Example 3.1 (Difficulty: 555-605 Level)
If f(x) = g(x - 1), where g(x) = |x| + 1, which of the following must be true?
(A) f(x) < 0
(B) f(x) = 0
(C) f(x) > 0
(D) f(x) = 2
(E) f(x) = g(x)
Question Discussion.
Example 3.2 (Difficulty: 555-605 Level)
The domain of the function \(f(x) = \frac{\sqrt{x - 1} }{x+1}\) is the set of all real numbers that are
A. greater than 1
B. greater than or equal to 1
C. not equal to -1
D. less than or equal to 1
E. less than 1
Question Discussion.
Example 3.3 (Difficulty: 655-705 Level)
The domain of the function \(f(x) = \sqrt{\sqrt{x+2} - \sqrt{4-x} }\) is the set of real numbers x such that
A. -2 ≤ x ≤ 1
B. -2 ≤ x ≤ 4
C. 1 ≤ x ≤ 2
D. 1 ≤ x ≤ 4
E. 2 ≤ x ≤ 4
Question Discussion.
Example 3.4 (Difficulty: 705-805 Level)
The range of a function is the set of all possible values of the function. If the function f is defined by \(f(x) = \frac{1}{x^2 + 1}\) for all real numbers x, the range of f is the set of
A. all real numbers
B. all real numbers except -1 and 1
C. all positive real numbers
D. all real numbers greater than or equal to 0 and less than or equal to 1
E. all positive real numbers less than or equal to 1
Question Discussion.
Pattern #4: Number Properties in Functions
- Involves rounding, greatest integer, or other number property logic applied within a function.
Example 4.1 (Difficulty: 505-555 Level)
For all numbers n, the symbol [n] denotes the greatest integer less than or equal to n. What is the value of [√2] + [√5] + [√7] ?
A. 3
B. 4
C. 5
D. 6
E. 8
Question Discussion.
Example 4.2 (Difficulty: 555-605 Level)
For all real numbers x, let \(\overline{x}\) denote the number obtained by rounding x to the nearest hundredth. If x = 21.9445, what is the value of \(10x−10\overline{x}\) ?
A. 0.005
B. 0.045
C, 0.095
D. 0.45
E. 0.95
Question Discussion.
Theory:
• How to Interpret Unfamiliar Symbols
• Functions on GMAT
• A Closer Look at GMAT Function Questions
Questions:
• Operations/functions defining algebraic/arithmetic expressions
• Rounding Functions Problems
• Symbols Representing Arithmetic Operation
• Various Functions Problems
• Hard Questions
• Medium Questions
• Easy Questions
23 Apr 2025, 06:45
Kudos
Bookmarks
The whole collection:
- GMAT Question Patterns: Work and Rate Problems
- GMAT Question Patterns: Arithmetic
- GMAT Question Patterns: Absolute Value
- GMAT Question Patterns: Distance and Speed
- GMAT Question Patterns: Remainders
- GMAT Question Patterns: Algebra
- GMAT Question Patterns: Overlapping Sets
- GMAT Question Patterns: Sequences
- GMAT Question Patterns: Functions and Custom Characters
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