8.1 Function Fundamentals, Notation, Domain, & Range

8.1 Function Fundamentals, Notation, Domain, & Range

Functions serve as the central organizing concept of high-school algebra and applied mathematical modeling across TABE 13&14 Mathematics Levels D and A. Whether calculating monthly utility billing rates, predicting machinery depreciation over time, or optimizing manufacturing output, functions provide a rigorous mathematical framework for describing how one changing quantity depends predictably on another.

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1. Mathematical Relations vs. Functions

A relation is simply any set of ordered pairs $(x, y)$ that establishes a connection between an input value and an output value. The set of all allowable input values ($x$-coordinates) is called the domain, while the set of all resulting output values ($y$-coordinates) is called the range.

A function is a specialized type of relation governed by a strict mathematical rule:

The "Vending Machine" Conceptual Model

Think of a function as a dependable vending machine:

• You press button B4 (the input $x$), and the machine dispenses Granola Bar (the single output $y$). That is a function.
• If pressing B4 sometimes gives you a granola bar and other times gives you a bottle of water, the machine is broken and unpredictable. That relation is not a function.
• However, if pressing button B4 gives you a granola bar, and pressing button C2 also gives you an identical granola bar, the machine is still operating predictably. Multiple different inputs can produce the same output, and the relation remains a valid function.

Function Validity Criterion:
  [Input: x1] ───> [Output: y1]   (Valid: One input to one output)
  [Input: x2] ───> [Output: y1]   (Valid: Different inputs to same output)
  
  [Input: x3] ──┬─> [Output: y2]  (INVALID: Single input producing
                └─> [Output: y3]            two different outputs -> NOT A FUNCTION)

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2. Four Common Representations of Functions

On the TABE assessment, relations are presented in four distinct mathematical formats:

A. Sets of Ordered Pairs

To determine if a set of ordered pairs is a function, inspect the first coordinates ($x$-values):

• Function: $\{(-3, 5), (0, 2), (4, 7), (9, 5)\}$ $\implies$ Every $x$-coordinate is unique ($-3, 0, 4, 9$). Notice that the $y$-value $5$ appears twice, which is completely permissible.
• Not a Function: $\{(2, 8), (5, 11), (2, -4), (7, 1)\}$ $\implies$ The input $x = 2$ maps to two different outputs ($y = 8$ and $y = -4$).

B. Mapping Diagrams

A mapping diagram uses two parallel ovals with arrows connecting domain elements to range elements:

C. Input-Output Tables

In tabular representations, scan the $x$-column for repeated entries with conflicting $y$-values:

D. Algebraic Equations

An equation represents $y$ as a function of $x$ if solving for $y$ yields a single expression:

• Linear equation: $y = 3x - 5$ $\implies$ For every $x$, there is exactly one $y$ (Function).
• Quadratic equation: $y = x^2 + 4$ $\implies$ Each $x$ produces one unique $y$ (Function).
• Circle equation: $x^2 + y^2 = 25 \implies y = \pm\sqrt{25 - x^2}$ $\implies$ An input like $x = 0$ yields two outputs ($y = +5$ and $y = -5$), so it is not a function.

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3. The Vertical Line Test on Coordinate Graphs

When a relation is graphed on the Cartesian coordinate plane, you can visually verify whether it is a function using the Vertical Line Test:

     Vertical Line Test Visualized:
     
     PARABOLA: y = x^2 (FUNCTION)             CIRCLE: x^2 + y^2 = 16 (NOT A FUNCTION)
             |                                         |
         \   |   /                                  .-----. 
          \  |  /                                 /    |    \
           \ | /                                 |     |     | <-- Vertical line intersects
      -------+------- x                     -----+-----+-----+--- x at TWO points:
             |                                   |     |     |    (0, 4) and (0, -4)
             |                                    \    |    /
     Vertical line crosses at ONCE                  '-----'

• Function Graphs: Non-vertical straight lines ($y = mx + b$), upward/downward opening parabolas ($y = ax^2 + bx + c$), exponential curves ($y = a \cdot b^x$), and square root curves ($y = \sqrt{x}$).
• Non-Function Graphs: Circles, ellipses, sideways opening parabolas ($x = ay^2$), and vertical lines ($x = c$).

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4. Function Notation $f(x)$ & Algebraic Evaluation

Function notation replaces the dependent variable $y$ with the symbol $f(x)$, read aloud as "f of x":

$$y = 4x - 7 \iff f(x) = 4x - 7$$

Evaluating Numeric Inputs

To evaluate $f(c)$, substitute the numerical constant $c$ in place of every instance of $x$ in the formula.

Worked Example 1: Given $f(x) = 2x^2 - 5x + 3$, evaluate $f(-3)$ and $f(4)$.

$$\begin{aligned} f(-3) &= 2(-3)^2 - 5(-3) + 3 = 2(9) + 15 + 3 = 18 + 15 + 3 = 36 \\ f(4) &= 2(4)^2 - 5(4) + 3 = 2(16) - 20 + 3 = 32 - 20 + 3 = 15 \end{aligned}$$

Evaluating Algebraic Expressions (Binomial Inputs)

When the input is itself an algebraic expression such as $(x + h)$ or $(x - 2)$, substitute the entire grouped expression into each variable position and simplify.

Worked Example 2: Given $g(x) = 3x^2 - 4x + 5$, find and simplify $g(x - 2)$.

$$\begin{aligned} g(x - 2) &= 3(x - 2)^2 - 4(x - 2) + 5 \\ &= 3(x^2 - 4x + 4) - 4(x - 2) + 5 \quad \text{(Expand squared binomial)} \\ &= (3x^2 - 12x + 12) - 4x + 8 + 5 \quad \text{(Distribute coefficients)} \\ &= 3x^2 + (-12x - 4x) + (12 + 8 + 5) \quad \text{(Combine like terms)} \\ &= 3x^2 - 16x + 25 \end{aligned}$$

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5. Domain and Range: Continuous vs. Discrete Sets

• Discrete Domain: Consists of separate, isolated values (e.g., the number of employee shifts worked $n \in \{1, 2, 3, 4\}$, tickets sold, or packages shipped). Represented as a list of numbers enclosed in set braces $\{x_1, x_2, \dots\}$.
• Continuous Domain: Consists of all unbroken real numbers within an interval (e.g., elapsed time, temperature, weight, distance). Represented using inequalities or interval notation.

Interval Notation Reference Guide

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6. Determining Restricted Algebraic Domains

In real-valued algebra, unless stated otherwise, the domain of a function is assumed to be all real numbers $(-\infty, \infty)$ except values that cause mathematically undefined operations. On the TABE assessment, watch for two primary domain restrictions:

Algebraic Domain Restrictions:
├── 1. Denominator Restriction ──> Denominator CANNOT equal 0  (Division by zero undefined)
└── 2. Radical Restriction ──────> Radicand under even root MUST be >= 0  (No imaginary numbers)

Restriction 1: Eliminating Division by Zero

Set each denominator equal to zero and solve; the domain excludes those specific values.

Worked Example 3: Find the domain of $f(x) = \frac{3x + 1}{2x - 8}$.

$$\begin{aligned} 2x - 8 &= 0 \implies 2x = 8 \implies x = 4 \\ \text{Domain:} &\quad x \neq 4 \quad \text{or in interval notation: } (-\infty, 4) \cup (4, \infty) \end{aligned}$$

Restriction 2: Non-Negative Radicands under Even Roots

Set the algebraic expression inside the radical (the radicand) greater than or equal to zero ($\ge 0$) and solve the inequality.

Worked Example 4: Find the domain of $h(x) = \sqrt{5x - 15}$.

$$\begin{aligned} 5x - 15 &\ge 0 \implies 5x \ge 15 \implies x \ge 3 \\ \text{Domain:} &\quad [3, \infty) \end{aligned}$$

Combined Restrictions: Radical in the Denominator

When a square root sits in the denominator, the radicand must be strictly positive ($> 0$), combining both rules (it cannot be negative, and it cannot equal zero).

Worked Example 5: Find the domain of $g(x) = \frac{\sqrt{2x - 6}}{x - 7}$.

1. Radicand condition (numerator): $2x - 6 \ge 0 \implies 2x \ge 6 \implies x \ge 3$.
2. Denominator condition: $x - 7 \neq 0 \implies x \neq 7$.
3. Combined Domain: All real numbers greater than or equal to $3$, excluding $7$. $$\text{Interval Notation: } [3, 7) \cup (7, \infty)$$