Comprehensive Study Guide: Section 1-1 Functions and Continuity

Relations, Domain, and Range

• Definition of a Relation:

  • A relation is any set of ordered pairs.

  • The ordered pairs in a relation do not need to form a linear pattern, create a recognizable picture, or be connected. They can simply be a collection of discrete points.

  • Relations are written using set notation with braces: $\{\dots\}$. If drawing set braces is difficult, a combination resembling an 'e' and a '3' stacked can be used as a substitute symbol.

  • Example Relation:         $\{ (4, -1), (0, 3), (-1, 2), (5, 2) \}$

• Definition of Domain:

  • The domain is the set of all input values in a relation, which corresponds to the $x$-coordinates.

  • When listing the domain of a relation:

    • The values must be enclosed in set notation braces $\{\dots\}$.

    • The values should be arranged in standard form, ordered from least to greatest.

    • Repeating values must not be listed more than once. The domain simply states which numbers are present in the set of inputs, regardless of how many times they appear.

  • Domain of the Example Relation:         $\{-1, 0, 4, 5\}$

• Definition of Range:

  • The range is the set of all output values in a relation, which corresponds to the $y$-coordinates.

  • Note on Terminology: The word "range" has three distinct definitions in mathematics, but in the context of functions, it refers strictly to the set of outputs.

  • When listing the range of a relation:

    • The values must be enclosed in set notation braces $\{\dots\}$.

    • The values should be listed in order from least to greatest.

    • Repeating values are listed only once.

  • Range of the Example Relation:         $\{-1, 2, 3\}$         (The output value $2$ appears twice in the ordered pairs $( -1, 2 )$ and $( 5, 2 )$, but it is only recorded once in the range set).

Functions and One-to-One Functions

• Universal Definition of a Function:

  • A function is a mathematical relation in which each element in the domain ($x$) is paired with exactly one element in the range ($y$).

• Shortcut Definition of a Function:

  • A relation is a function if there are no repeating values in the domain ($x$).

  • Explanation of Repeating Inputs: Consider the ordered pairs $(2, 3)$ and $(2, 1)$. Here, the domain value $2$ is paired with two distinct range values ($3$ and $1$). Because the input $2$ repeats with different outputs, the relation violates the definition of a function.

  • Evaluation of Example Relation $\{(4, -1), (0, 3), (-1, 2), (5, 2)\}$:

    • Domain elements: $4, 0, -1, 5$.

    • None of the domain values repeat.

    • Therefore, this relation is a function.

    • Explanation format for tests: "Yes, it is a function because there are no repeating values in the domain."

• Definition of a One-to-One ($1\text{-to-}1$) Function:

  • A one-to-one function is a specific type of function where each element of the function's range ($y$) is paired with exactly one element in the domain ($x$).

  • Shortcut Definition: A relation is a one-to-one function if there are no repeating values in the domain AND no repeating values in the range.

  • Evaluation of Example Relation $\{(4, -1), (0, 3), (-1, 2), (5, 2)\}$:

    • Range elements: $-1, 3, 2, 2$.

    • The range element $2$ repeats (paired with both $-1$ and $5$).

    • Therefore, the relation is not a one-to-one function.

    • Explanation format for tests: "No, it is not one-to-one because there is a repeating value in the range."

  • Application: One-to-one functions become primary focal points when studying inverse functions.

Graphical Tests for Functions

• Vertical Line Test (VLT):

  • A graphical method used to determine whether a graph represents a function.

  • Rule: If any vertical line drawn through a graph intersects/touches the graph more than once, the graph is not a function.

  • Mathematical Reason: If a vertical line touches a graph at multiple points, those points share the exact same $x$-coordinate but have different $y$-coordinates, indicating a repeated domain value.

• Horizontal Line Test (HLT):

  • A graphical method used to determine whether a function is one-to-one ($1\text{-to-}1$).

  • Rule: A graph represents a one-to-one function if and only if it passes both the Vertical Line Test and the Horizontal Line Test.

  • Mathematical Reason: Intersections with a horizontal line represent repeated $y$-values (repeaters in the range).

• Graphical Analysis Example — Parabola:

  • Shape: Parabola (quadratic function).

  • Vertical Line Test: A vertical line dragged across the parabola touches it at most once everywhere. It passes the Vertical Line Test.

  • Horizontal Line Test: A horizontal line intersects the parabola twice. It fails the Horizontal Line Test.

  • Conclusion: The parabola is a function, but it is not a one-to-one function.

Notations for Domain and Range

When determining domain and range from a graph:

• Domain ($x$): Analyzed strictly from left to right (minimum $x$-value to maximum $x$-value).

• Range ($y$): Analyzed strictly from bottom to top (minimum $y$-value to maximum $y$-value).

• Three Standard Notations:

  1. Set Notation: Uses standard inequality symbols ($<, >, \le, \ge$) to define boundaries (e.g., $x \ge 2$).

  2. Set-Builder Notation: A formal set notation that specifies the variable and condition using a vertical bar ($\mid$), read as "such that."

     • Structure: $\{ x \mid x \ge 2 \}$ (read as "the set of all $x$ such that $x$ is greater than or equal to $2$").

  3. Interval Notation: Advanced notation used for continuous sets of real numbers.

     • Brackets [ or ]: Indicate closed endpoints (inclusive values where the graph sits directly on a point, equivalent to a closed circle).

     • Parentheses ( or ): Indicate open endpoints (exclusive values, open circles, or boundary values).

     • Infinities ($\infty$ or $-\infty$): ALWAYS take parentheses ( or ) because a graph can never reach infinity.

     • Structure: Written in left-to-right or bottom-to-top order as [minimum, maximum).

• Detailed Graph Example 1 — V-Shape Ray / Absolute Value Graph:

  • Vertex Location: $(2, 1)$

  • Domain Analysis ($x$-axis, left to right):

    • Leftmost boundary starts at $x = 2$ and extends infinitely to the right ($\infty$).

    • Set Notation:             $x \ge 2$

    • Set-Builder Notation:             $\{ x \mid x \ge 2 \}$

    • Interval Notation:             $[2, \infty)$

  • Range Analysis ($y$-axis, bottom to top):

    • The graph extends infinitely downwards ($-\infty$) and infinitely upwards ($\infty$).

    • Set Notation:             $-\infty < y < \infty$             (Alternatively represented as $y \in \mathbb{R}$, where $\in$ means "element of" and $\mathbb{R}$ represents the set of all real numbers).

    • Set-Builder Notation:             $\{ y \mid y \in \mathbb{R} \}$

    • Interval Notation:             $(-\infty, \infty)$

  • Function and One-to-One Status:

    • Fails the Vertical Line Test if a vertical line intersects both upper and lower sections; thus, it is neither a function nor one-to-one.

• Summary of Special Lines:

  • Diagonal Lines: Domain is $(-\infty, \infty)$; Range is $(-\infty, \infty)$ (All real numbers).

  • Vertical Lines: Domain is $x = c$ (a single value); Range is $(-\infty, \infty)$. Fails VLT (not a function).

  • Horizontal Lines: Domain is $(-\infty, \infty)$; Range is $y = c$ (a single value). Passes VLT (is a function, but not one-to-one).

Types of Functions: Continuity and Discontinuity

• Continuous Function:

  • A function whose graph is a single line or unbroken curve.

  • You can trace the entire graph from end to end without lifting your pencil. Continuous graphs typically end with arrows.

• Discontinuous Function:

  • A function whose graph contains breaks, gaps, jumps, or holes, requiring you to lift your pencil to draw it.

• Discrete Function:

  • A specific category of discontinuous function where the graph consists entirely of isolated, unconnected points (dots).

• Determining Continuity in Real-World Contexts (The Breakdown Rule):

  • To determine if a real-world scenario is continuous, discrete, or neither, test whether the variables can be divided into fractional parts/pieces:

    • Continuous: BOTH the domain ($x$) and range ($y$) can be meaningfully broken down into fractional parts.

    • Discrete: BOTH variables consist of discrete, individual countable items (e.g., number of living people cannot be split into fractional parts).

    • Neither: ONE variable can be broken down into fractional parts while the OTHER variable cannot.

• Examples of Context Variables:

  • Count of People: Cannot have half a person; represents a discrete variable.

  • Person's Height: Can be measured in fractional feet or inches ($5.5\, \text{ft}$, $0.25\, \text{in}$); represents a continuous variable.

Real-World Context Examples

• Example 2 — U.S. House of Representatives Data:

  • Data Set:

    • California (CA): Population = $37.25\text{ million}$, Representatives = $53$

    • Texas (TX): Population = $25.15\text{ million}$, Representatives = $36$

    • New York (NY): Population = $19.38\text{ million}$, Representatives = $27$

    • Florida (FL): Population = $18.80\text{ million}$, Representatives = $27$

    • Illinois (IL): Population = $12.83\text{ million}$ ($12,830,000$), Representatives = $18$

  • Variable Classification:

    • Independent Variable ($x$ / Domain): Population in millions (number of representatives depends on population).

    • Dependent Variable ($y$ / Range): Number of representatives.

  • Domain (List in set notation, ordered least to greatest):         $\{ 12.83, 18.8, 19.38, 25.15, 37.25 \}$

  • Range (List in set notation, ordered least to greatest):         $\{ 18, 27, 36, 53 \}$         (The output value $27$ is listed only once despite appearing for both New York and Florida).

  • One-to-One Evaluation:

    • Is it one-to-one? No, because there is a repeating value in the range ($27$ appears twice).

  • Continuity Classification:

    • Population ($x$) can be divided into fractional parts (e.g., $37.25\text{ million}$).

    • Representatives ($y$) cannot be divided into fractional parts (you cannot have half a representative).

    • Classification: Neither continuous nor discrete.

• Example 3 — Maru's Art and Design Services:

  • Pricing Structure:

    • $\$150$ flat fee for up to $2\text{ hours}$ of work ($0 < x \le 2$).

    • $\$250$ flat fee for $2$ or more hours, but less than $4\text{ hours}$ ($2 \le x < 4$).

    • $\$300$ flat fee for $4$ or more hours, but less than $6\text{ hours}$ ($4 \le x < 6$).

    • No set rate defined for $6\text{ or more hours}$.

  • Variable Classification:

    • Independent Variable ($x$): Number of hours worked.

    • Dependent Variable ($y$): Total fee in dollars.

  • Continuity Classification:

    • Hours ($x$) can be broken into fractional parts (e.g., half an hour, 45 minutes).

    • Fee ($y$) in this flat-fee model cannot be broken into parts (the client pays fixed tier amounts of $\$150$, $\$250$, or $\$300$ without intermediate hourly breakdown).

    • Classification: Neither continuous nor discrete.

  • Domain Analysis:

    • Hours extend from greater than $0$ up to less than 6$.\n * Interval Notation: (0, 6)\n * *(Note: 0$is open `(` because hiring a designer implies working strictly greater than$0$hours;$6 is open `)` because rates for 6 or more hours are undefined).*\n * *Range Analysis:*\n * The fee consists only of three discrete amounts.\n * Set Notation: { 150, 250, 300 }\n\n# Piecewise Functions and Advanced Graphical Domain & Range\n\n* **Detailed Graph Example 4 — Discontinuous Piecewise Ray Graph:**\n * *Component 1:* Open circle at (-1, 1) with a ray extending up and to the left.\n * *Component 2:* Closed circle at (0, -2) with a ray extending down and to the right.\n * **Domain Analysis (x-axis, left to right):**\n * Left ray extends to -\infty$on the left and terminates at$x = -1$with an open circle:$(-\infty, -1).\n * Right ray begins at x = 0$with a closed circle and extends rightward to$\infty$:$[0, \infty).\n * *Gap Test:* There is a gap along the x$axis between$x = -1$and$x = 0$where no graph exists. Disjoint intervals are connected using the Union symbol ($\cup).\n * *Domain in Interval Notation:*\n            (-\infty, -1) \cup [0, \infty)\n * *(Note: If two rays overlap along the x$-axis without a gap, the domain combines into a single continuous interval like$(-\infty, \infty)).*\n * **Range Analysis (y-axis, bottom to top):**\n * Bottom ray extends downward to -\infty$and reaches a top endpoint at$y = -2$with a closed circle:$(-\infty, -2].\n * Top ray begins at y = 1$with an open circle and extends upward to$\infty$:$(1, \infty).\n * *Gap Test:* There is a vertical gap along the y$axis between$y = -2$and$y = 1.\n * *Range in Interval Notation:*\n            (-\infty, -2] \cup (1, \infty)\n\n# Homework Assignment and Guidelines\n\n* **Assignment Details:**\n * **Source:** Workbook Page 11\n * **Problems:** 1 \text{ through } 29 \text{ odd}$$

• Homework Rules and Protocol:

  • Do NOT tear pages out of the workbook. Workbooks contain perforated pages, but pages must remain inside because materials may be reused.

  • Complete all homework on separate notebook paper.

  • Word Problems: Do not copy down the narrative text of word problems.

  • Graphs: Do not redraw or copy graphs when solving domain and range problems.

  • Standard Math Problems: Write down the original problem and show all work step-by-step.

  • Paper Header Requirements: Top of the paper must clearly state:

    1. Student Name

    2. First Period

    3. Assignment Identification (e.g., "Page 11 #1-29 odd")