Functions, Domain, and Range

Section 1: Relations vs. Functions

• Relation: A set of ordered pairs, or any rule that connects an independent variable (typically $x$) with a dependent variable (typically $y$).

• Function: A specific type of relation in which each element of the independent variable (domain / $x$) corresponds to exactly one element of the dependent variable (range / $y$).

• Fundamental Principle: All functions are relations, but not all relations are functions. For a relation to qualify as a function, there must be only one $y$-value associated with any given $x$-value.

Investigation: Comparing $y = x^2$ and $x = y^2$

• Table of Values for $y = x^2$:

• Table of Values for $x = y^2$:

• Graphical Comparison and Vertical Line Analysis:

  • Graphing $y = x^2$ produces an upward-opening parabola centered at $(0, 0)$. Drawing vertical lines across the graph (such as $x = -2$, $x = -1$, $x = 0$, $x = 1$, and $x = 2$) shows that each vertical line intersects the curve at exactly one point. Therefore, $y = x^2$ is a function.

  • Graphing $x = y^2$ produces a right-opening parabola centered at $(0, 0)$. Drawing vertical lines across the graph (such as $x = 1$ or $x = 4$) shows that vertical lines intersect the curve at two distinct points (for instance, at $(1, 1)$ and $(1, -1)$, or $(4, 2)$ and $(4, -2)$). Because a single $x$-value maps to multiple $y$-values, $x = y^2$ is not a function (it is a relation only).

Section 2: The Vertical Line Test

• Definition: The Vertical Line Test is a visual method for determining whether a relation represented graphically is a function. If every possible vertical line intersects the graph of the relation at at most one point, then the relation is a function. If any vertical line intersects the graph at two or more points, the relation is not a function.

Example 1: Determining Function Status from Graphs

• a) Linear Function: $y = -4x + 5$

  • Graph Characteristics: A line with a $y$-intercept of $(0, 5)$ and a slope of $-4$.

  • Vertical Line Test: Every vertical line intersects the graph at exactly one point.

  • Conclusion: Function.

• b) Horizontal Parabola: $x = (y - 2)^2 + 2$

  • Graph Characteristics: A horizontal parabola opening to the right with vertex at $(2, 2)$.

  • Vertical Line Test: A vertical line at $x = 6$ intersects the graph at two points: $(6, 0)$ and $(6, 4)$.

  • Conclusion: Not a function.

• c) Vertical Parabola: $y = -(x + 4)^2 + 3$

  • Graph Characteristics: A parabola opening downward with vertex at $(-4, 3)$.

  • Vertical Line Test: Every vertical line intersects the graph at exactly one point.

  • Conclusion: Function.

• d) Circle: $(x - 1)^2 + (y + 1)^2 = 16$

  • Graph Characteristics: A circle centered at $(1, -1)$ with a radius of 4$.\n - **Vertical Line Test**: A vertical line at x = 1$intersects the graph at two points:$(1, 3)$and$(1, -5).\n - **Conclusion**: **Not a function**.\n\n# Section 3: Domain and Range\n\n- **Domain (D$)**: For any relation, the set of all values of the independent variable (typically$x) for which the relation is defined.\n- **Range (R$)**: The set of corresponding values of the dependent variable (typically$y) generated by the relation.\n- **Real Numbers (\\mathbb{R}$)**: The set of all numbers including integers, terminating decimals, repeating decimals, non-terminating decimals, and non-repeating decimals. It is represented by the symbol$\mathbb{R}.\n- **Function Mapping Rule**: For a relation to be a function, each element in the domain must map to **exactly one element** in the range.\n\n## General Notation for Domain and Range\n\n- **Continuous Sets (Set-Builder Notation)**:\n - Domain: D = \{x \in \mathbb{R} \mid \text{restrictions}\ ull\}\n - Range: R = \{y \in \mathbb{R} \mid \text{restrictions}\ ull\}\n- **Discrete Sets (Roster Notation)**:\n - Domain: D = \{x_1, x_2, x_3, \dots\}\n - Range: R = \{y_1, y_2, y_3, \dots\}\n\n## Example 2: Discrete Data Sets\n\n- **a) Ordered Pairs Set**: \ ull\{(-3, 4), (5, -6), (-2, 7), (5, 3), (6, -8)\ ull\}\n - **Domain**: D = \{-3, -2, 5, 6\}\n - **Range**: R = \{-8, -6, 3, 4, 7\}\n - **Function Status**: **Not a function**, because the input x = 5$produces two distinct outputs ($y = -6$and$y = 3).\n\n- **b) Table of Data (Age vs. Number)**:\n\n| Age (x$) | Number ($y) |\n| :---: | :---: |\n| 4$|$8 |\n| 5$|$12 |\n| 6$|$5 |\n| 7$|$22 |\n| 8$|$14 |\n| 9$|$9 |\n| 10$|$11 |\n\n - **Domain**: D = \{4, 5, 6, 7, 8, 9, 10\}\n - **Range**: R = \{5, 8, 9, 11, 12, 14, 22\}\n - **Function Status**: **Function**, because each age (x$) pairs with exactly one number ($y).\n\n## Example 3: Continuous Relations\n\n- **a) Linear Function**: y = 2x - 5\n - **Graph**: A straight line with slope 2$and$y$-intercept$(0, -5).\n - **Domain**: D = \{x \in \mathbb{R}\ ull\}\n - **Range**: R = \{y \in \mathbb{R}\ ull\}\n - **Function Status**: **Function**\n\n- **b) Upward Parabola**: y = (x - 1)^2 + 3\n - **Graph**: Parabola opening upward with vertex at (1, 3).\n - **Domain**: D = \{x \in \mathbb{R}\ ull\}\n - **Range**: R = \{y \in \mathbb{R} \mid y \ge 3\ ull\}\n - **Function Status**: **Function**\n\n- **c) Square Root Function**: y = \sqrt{x - 1} + 3\n - **Graph**: Half-parabola starting at endpoint (1, 3) extending rightward and upward.\n - **Algebraic Derivation**:\n - Under the radical: x - 1 \ge 0 \implies x \ge 1.\n - Radical value: \sqrt{x - 1} \ge 0 \implies y \ge 3$.

  • Domain: $D = \\{x \\in \\mathbb{R} \\mid x \\ge 1\\null\\}$

  • Range: $R = \\{y \\in \\mathbb{R} \\mid y \\ge 3\\null\\}$

  • Function Status: Function

• d) Circle: $x^2 + y^2 = 36$

  • Graph: Circle centered at $(0, 0)$ with radius r = 6$.\n - **Domain**: D = \{x \in \mathbb{R} \mid -6 \le x \le 6\ ull\}\n - **Range**: R = \{y \in \mathbb{R} \mid -6 \le y \le 6\ ull\}\n - **Function Status**: **Not a function**\n\n# Section 4: Rational Functions and Asymptotes\n\n- **e) Rational Function**: y = \frac{1}{x + 3}\n\n\n\n- **Definition of Asymptote**: A line that a curve approaches arbitrarily closely as it heads toward infinity or negative infinity.\n\n- **Vertical Asymptote (x = -3)**:\n - **Explanation**: Division by zero is undefined. Consequently, the denominator of the function cannot equal zero.\n - **Algebraic Determination**: Set denominator equal to zero: x + 3 = 0 \implies x = -3$.

  • Restriction: x \\neq -3$.\n - **Asymptote Line**: The vertical line x = -3 is a vertical asymptote.\n\n- **Horizontal Asymptote (y = 0)**:\n - **Explanation**: A fraction with a constant non-zero numerator (1) can never equal zero.\n - **General Rule**: For all rational functions where the degree of the denominator is higher than the degree of the numerator, there is a horizontal asymptote at y = 0$.

  • Restriction: y \\neq 0$.\n - **Asymptote Line**: The horizontal line y = 0$(the$x-axis) is a horizontal asymptote.\n\n- **Domain**: D = \{x \in \mathbb{R} \mid x \ eq -3\ ull\}\n- **Range**: R = \{y \in \mathbb{R} \mid y \ eq 0\ ull\}$$

• Function Status: Function