8.1 Function Fundamentals, Notation, Domain, & Range
Key Takeaways
- A mathematical relation is any set of ordered pairs $(x, y)$; a relation is a function if and only if each unique input ($x$-value from the domain) maps to exactly one output ($y$-value in the range).
- Functions can be represented through ordered pairs, mapping diagrams, input-output tables, algebraic equations, and graphs; on a coordinate plane, the Vertical Line Test verifies that no vertical line crosses a function graph more than once.
- Function notation $f(x)$ represents the dependent output value generated by rule $f$ for an independent input $x$; evaluating functions requires direct algebraic substitution of numerical constants (such as $f(3)$) or polynomial expressions (such as $f(x + 2)$).
- The domain represents all valid independent inputs ($x$) and the range represents all resulting dependent outputs ($y$); algebraic domains are restricted by eliminating values that produce division by zero ($x \neq 0$) or negative numbers under even radicals (radicand $\ge 0$).
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.
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:
[!IMPORTANT] Formal Definition of a Function: A relation is a function if and only if each input element ($x$) in the domain corresponds to exactly one output element ($y$) in the range.
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)
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:
| Mapping Type | Arrow Pattern | Function Status |
|---|---|---|
| One-to-One | Exactly one arrow leaves each domain element and points to a unique range element | Function |
| Many-to-One | Distinct domain elements have arrows pointing to the same range element | Function |
| One-to-Many | Two or more arrows branch out from a single domain element | NOT a Function |
C. Input-Output Tables
In tabular representations, scan the $x$-column for repeated entries with conflicting $y$-values:
| Table 1 (Function) | $x$ | $y$ |
|---|---|---|
| $-2$ | $4$ | |
| $0$ | $0$ | |
| $2$ | $4$ | |
| $4$ | $16$ |
| Table 2 (Not a Function) | $x$ | $y$ |
|---|---|---|
| $3$ | $12$ | |
| $5$ | $20$ | |
| $3$ | $18$ | |
| $8$ | $32$ |
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.
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:
[!TIP] The Vertical Line Test: If every vertical line that can be drawn through the coordinate plane intersects a graph at most once, the graph represents a function. If any vertical line intersects the graph at two or more points, the graph is not a function.
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 ($x = c$).
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":
[!CAUTION] Common Notation Trap: The notation $f(x)$ does NOT mean $f$ multiplied by $x$. It signifies that $f$ is the name of the function rule, and $x$ is the input variable being operated upon.
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 $f(4)$.
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}$$ --- ## 5. Domain and Range: Continuous vs. Discrete Sets - **Discrete Domain:** Consists of separate, isolated values (e.g., the number of employee shifts worked $n \in \{1, 2, 3, 4\}$, tickets sold, or packages shipped). Represented as a list of numbers enclosed in set braces $\{x_1, x_2, \dots\}$. - **Continuous Domain:** Consists of all unbroken real numbers within an interval (e.g., elapsed time, temperature, weight, distance). Represented using inequalities or interval notation. ### Interval Notation Reference Guide | Inequality Statement | Interval Notation | Number Line Endpoint Style | | :--- | :---: | :--- | | $a \le x \le b$ | $[a, b]$ | Solid / Closed circles at $a$ and $b$ (included) | | $a < x < b$ | $(a, b)$ | Open circles at $a$ and $b$ (excluded) | | $x \ge a$ | $[a, \infty)$ | Solid circle at $a$, arrow extending right to positive infinity | | $x < b$ | $(-\infty, b)$ | Open circle at $b$, arrow extending left to negative infinity | | All Real Numbers | $(-\infty, \infty)$ | Entire number line shaded with arrows in both directions | --- ## 6. Determining Restricted Algebraic Domains In real-valued algebra, unless stated otherwise, the domain of a function is assumed to be all real numbers $(-\infty, \infty)$ **except** values that cause mathematically undefined operations. On the TABE assessment, watch for two primary domain restrictions: ```text Algebraic Domain Restrictions: ├── 1. Denominator Restriction ──> Denominator CANNOT equal 0 (Division by zero undefined) └── 2. Radical Restriction ──────> Radicand under even root MUST be >= 0 (No imaginary numbers) ``` ### Restriction 1: Eliminating Division by Zero Set each denominator equal to zero and solve; the domain excludes those specific values. **Worked Example 3:** Find the domain of $f(x) = \frac{3x + 1}{2x - 8}$. $$\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)$$Which of the following sets of ordered pairs represents a valid mathematical function?
Given the quadratic function f(x) = 3x^2 - 4x + 5, what is the simplified algebraic expression for f(x - 2)?
What is the domain of the function g(x) = √(2x - 6) / (x - 7)?