4.4 Functions, Defined Operational Symbols, and Sequences

4.4 Functions, Defined Operational Symbols, and Sequences

Quick Summary: Functions, custom operators, and sequences test your algorithmic discipline and pattern-recognition skills. Success requires adhering to strict inside-out evaluation rules for composite functions, treating custom symbols as literal algebraic templates, and mastering the core closed-form formulas for arithmetic and geometric progressions.

The GMAT Focus Edition Quantitative Reasoning section frequently assesses how reliably you can apply novel rules, decode iterative relationships, and project sequences into large index values without manual enumeration. Whether encountering an invented operator symbol or a recursive sequence defined across multiple steps, your approach must remain structured, precise, and fast.

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Algebraic Functions and Composite Mappings

In standard mathematics, a function $f$ is a rule that assigns to each input value $x$ in its domain exactly one output value $f(x)$ in its range.

Domain Restrictions on the GMAT

Because the GMAT operates exclusively within the real number system, domain restrictions arise from two primary mathematical boundaries:

1. Denominators Cannot Equal Zero: In $f(x) = \frac{2x + 1}{x - 3}$, the domain includes all real numbers except $x = 3$.
2. Even Radicands Must Be Non-Negative: In $g(x) = \sqrt{2x - 8}$, the expression under the square root must satisfy $2x - 8 \ge 0 \implies x \ge 4$.

Composite Functions: The Inside-Out Evaluation Rule

A composite function represents the application of one function to the result of another, written as $f(g(x))$ or $(f \circ g)(x)$.

The Inside-Out Rule: Always compute the value of the innermost function first, then pass that output as the input to the outer function.

Example: Let $f(x) = 2x^2 - 3x + 1$ and $g(x) = x + 4$. Find $f(g(-2))$.

1. Inner Function: Calculate $g(-2)$: $g(-2) = -2 + 4 = 2$
2. Outer Function: Substitute $2$ into $f(x)$: $f(2) = 2(2^2) - 3(2) + 1 = 2(4) - 6 + 1 = 8 - 6 + 1 = 3$ Therefore, $f(g(-2)) = 3$.

Important Non-Commutativity Note: In general, $f(g(x)) \neq g(f(x))$. Evaluating $g(f(-2))$ for the functions above yields: $f(-2) = 2(-2)^2 - 3(-2) + 1 = 8 + 6 + 1 = 15 \implies g(15) = 15 + 4 = 19 \neq 3$

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Defined Operational Symbols ("Strange Symbols")

The GMAT frequently invents novel mathematical symbols—such as $\star, #, \diamond, \odot, \oplus, \Delta,$ or $&$—to evaluate your capacity to follow defined operations without preconceived assumptions.

The Template Substitution Strategy

Never allow an unfamiliar symbol to intimidate you. Treat the definition as a structural placeholder template. Replace the variables with empty boxes: $\text{If } a \star b = 2a - b^2, \quad \text{think: } [\text{First}] \star [\text{Second}] = 2[\text{First}] - [\text{Second}]^2$

Nested Custom Operators

When multiple custom operators are chained together, adhere strictly to parentheses hierarchy (evaluating innermost parentheses first):

Example: Let $x \odot y = \frac{xy}{x + y}$. Evaluate $(4 \odot 4) \odot 2$.

1. Evaluate Parentheses First: $4 \odot 4 = \frac{4 \times 4}{4 + 4} = \frac{16}{8} = 2$
2. Evaluate the Outer Operation: $2 \odot 2 = \frac{2 \times 2}{2 + 2} = \frac{4}{4} = 1$ The result is 1.

Testing Algebraic Properties of Custom Operators

GMAT questions may ask whether a custom operator obeys standard algebraic laws:

• Commutative Property: Does $a \star b = b \star a$? (Does changing the order of operands preserve the result?)
  • For $a \star b = a + b + ab$: $b \star a = b + a + ba = a + b + ab$. Commutative!
  • For $a \star b = 2a - b$: $b \star a = 2b - a \neq 2a - b$. Not commutative!
• Associative Property: Does $(a \star b) \star c = a \star (b \star c)$?
  • Must test explicitly; most custom operations involving powers or subtractions fail associativity.

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Arithmetic Sequences and Series

An arithmetic sequence (or arithmetic progression) is a sequence of numbers in which the difference between consecutive terms is constant. This constant is called the common difference ($d$):

$d = a_n - a_{n-1}$

The Three Essential Arithmetic Sequence Formulas

The Average and Median of Evenly Spaced Sets

In any arithmetic sequence (an evenly spaced set):

1. The mean (average) equals the median.
2. The mean is simply the average of the first and last terms: $\text{Mean} = \frac{a_1 + a_n}{2}$.
3. The Sum of the terms is always: $\text{Sum} = \text{Number of Terms} \times \text{Mean}$.

Worked Walkthrough: Divisibility Series Sum

Find the sum of all three-digit positive integers that are multiples of 7.

• First Term ($a_1$): Smallest 3-digit multiple of 7. $100 / 7 \approx 14.28 \implies 7 \times 15 = 105$.
• Last Term ($a_n$): Largest 3-digit multiple of 7. $999 / 7 \approx 142.71 \implies 7 \times 142 = 994$.
• Common Difference ($d$): $d = 7$.
• Number of Terms ($n$): $n = \frac{994 - 105}{7} + 1 = \frac{889}{7} + 1 = 127 + 1 = 128$
• Sum ($S_{128}$): $S_{128} = \frac{128}{2} \times (105 + 994) = 64 \times 1,099 = 64 \times (1,100 - 1) = 70,400 - 64 = 70,336$

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Geometric Sequences and Series

A geometric sequence is a sequence in which each term after the first is found by multiplying the preceding term by a non-zero constant called the common ratio ($r$):

$r = \frac{a_n}{a_{n-1}}$

The Core Geometric Sequence Formulas

• $n$-th Term Formula: $a_n = a_1 \cdot r^{n-1}$
• Sum of the First $n$ Terms ($S_n$ for $r \neq 1$): $S_n = a_1 \left(\frac{1 - r^n}{1 - r}\right) = a_1 \left(\frac{r^n - 1}{r - 1}\right)$

Alternating Geometric Sequences

If the common ratio is negative ($r < 0$), the sequence oscillates in sign: $\text{If } a_1 = 3 \text{ and } r = -2: \quad 3, -6, 12, -24, 48, -96, \dots$

• All odd-indexed terms ($a_1, a_3, a_5, \dots$) have the same sign as $a_1$ (positive).
• All even-indexed terms ($a_2, a_4, a_6, \dots$) have the opposite sign (negative).

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Recursive Sequences and Pattern Detection

A recursive sequence defines each term as a function of one or more preceding terms (e.g., $a_n = 2a_{n-1} + 3$, or the Fibonacci sequence $F_n = F_{n-1} + F_{n-2}$).

When a GMAT question asks for a distant term in a recursive sequence (such as $a_{75}$ or $a_{100}$), GMAC never expects you to perform 100 manual calculations. A repeating pattern or cycle is guaranteed to exist.

The Periodicity Protocol

1. Calculate the first 4 to 6 terms manually.
2. Identify the cycle length $k$ (the period after which values repeat).
3. Express the requested index $N$ in terms of the period using division with remainder: $N = qk + R$.
4. The $N$-th term matches the $R$-th term in the cycle ($a_N = a_R$). If the remainder $R = 0$, $a_N$ is the final term of the cycle ($a_k$).

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Step-by-Step Worked Problem Solving Examples

Example 1: Function Composition and Inversion

Problem: Let $f(x) = \frac{x + 2}{x - 1}$ for all real $x \neq 1$. What is the value of $f(f(3))$?

Step-by-Step Solution:

1. Evaluate the Inner Function $f(3)$: $f(3) = \frac{3 + 2}{3 - 1} = \frac{5}{2} = 2.5$
2. Pass Result to the Outer Function $f(2.5)$: $f\left(\frac{5}{2}\right) = \frac{\frac{5}{2} + 2}{\frac{5}{2} - 1}$
3. Simplify the Complex Fraction: $\text{Numerator: } \frac{5}{2} + \frac{4}{2} = \frac{9}{2}$ $\text{Denominator: } \frac{5}{2} - \frac{2}{2} = \frac{3}{2}$ $f\left(\frac{5}{2}\right) = \frac{9/2}{3/2} = \frac{9}{3} = 3$
4. Conclusion: $f(f(3)) = 3$. (Notice that $f(x)$ is an involution, meaning $f(f(x)) = x$ for all $x$ in its domain!).

Example 2: Arithmetic Sequence Sum with Unknown Bound

Problem: The first term of an arithmetic sequence is 7, and the 20th term is 83. What is the sum of the first 20 terms of this sequence?

Step-by-Step Solution:

1. Identify Given Data: First term $a_1 = 7$, 20th term $a_{20} = 83$, number of terms $n = 20$.
2. Select the Direct Sum Formula: Since the first and last terms are already known, deploy $S_n = \frac{n}{2}(a_1 + a_n)$ without needing to compute the common difference $d$: $S_{20} = \frac{20}{2}(a_1 + a_{20})$
3. Execute the Arithmetic: $S_{20} = 10 \times (7 + 83) = 10 \times 90 = 900$
4. Check via Common Difference (Verification): $a_{20} = a_1 + 19d \implies 83 = 7 + 19d \implies 19d = 76 \implies d = 4$ $S_{20} = \frac{20}{2}[2(7) + 19(4)] = 10[14 + 76] = 10(90) = 900$. Confirmed!

Example 3: Periodic Recursive Sequence

Problem: A sequence is defined by $a_1 = 3$ and $a_{n+1} = \frac{a_n - 1}{a_n + 1}$ for all integers $n \ge 1$. What is the value of $a_{75}$?

Step-by-Step Solution:

1. Generate the First Few Terms to Detect Periodicity:
   • $a_1 = 3$
   • $a_2 = \frac{3 - 1}{3 + 1} = \frac{2}{4} = \frac{1}{2}$
   • $a_3 = \frac{1/2 - 1}{1/2 + 1} = \frac{-1/2}{3/2} = -\frac{1}{3}$
   • $a_4 = \frac{-1/3 - 1}{-1/3 + 1} = \frac{-4/3}{2/3} = -2$
   • $a_5 = \frac{-2 - 1}{-2 + 1} = \frac{-3}{-1} = 3$
2. Identify the Cycle Length: Since $a_5 = a_1 = 3$, the sequence repeats every 4 terms: $\text{Cycle: } \left\{3, \frac{1}{2}, -\frac{1}{3}, -2\right\} \quad \implies \quad \text{Period } k = 4$
3. Apply Modular Arithmetic to Index 75: $75 \div 4 = 18 \quad \text{with a remainder of } 3 \quad (75 = 18 \times 4 + 3)$
4. Retrieve Corresponding Cycle Term: $a_{75} = a_3 = -\frac{1}{3}$

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High-Frequency GMAT Traps & Pacing Strategies

• Trap 1: The Off-by-One Fencepost Error. Forgetting to add 1 when counting terms in an arithmetic progression. The number of terms between 10 and 50 inclusive is $(50 - 10) + 1 = 41$, not 40.
• Trap 2: Assuming Custom Operators are Commutative. In $a \star b = 2a - b$, switching the inputs gives $b \star a = 2b - a$. Never rearrange operands around an unfamiliar symbol.
• Trap 3: Evaluating Composites Outside-In. Attempting to evaluate $f(g(x))$ by substituting $f$ into $g$ or multiplying $f(x)$ by $g(x)$. Always calculate the inner argument $g(x)$ first.
• Trap 4: Missing Alternating Signs in Geometric Sequences. Forgetting that if $r < 0$, terms oscillate between positive and negative values, meaning the sum or term values cannot be analyzed assuming monotonic growth.