Functions, Domain, Range, and Function Evaluation

Core Concept of Functions, Domain, and Range

• Definition of Function: A function is a mathematical relation or process that maps inputs to outputs such that every input corresponds to exactly one output.
• Domain: The set of all starting input values (the set in which mapping arrows originate, or the set of all $x$-values).
• Range: The set of all resulting output values (the set in which mapping arrows end, or the set of all $y$-values).
• The Universal Function Criterion: For any correspondence between two sets to be classified as a function, every member of the domain must correspond to exactly one member of the range.

Non-Function vs. Function Correspondences

• Course-to-Topic Relation (Non-Function Example):

  • Domain: Courses $\{\text{African History}, \text{Algebra}\}$
  • Range: Topics $\{\text{Ancient Civilization}, \text{Oral Traditions}, \text{Independence Movement}, \text{Equations}, \text{Graphs}\}$
  • Directional Mapping from Domain:
  • African History $\rightarrow$ Ancient Civilization
  • African History $\rightarrow$ Oral Traditions
  • African History $\rightarrow$ Independence Movement
  • Algebra $\rightarrow$ Equations
  • Algebra $\rightarrow$ Graphs
  • Evaluation: Inputting the single domain element "African History" into the machine yields $3$ separate outputs (Ancient Civilization, Oral Traditions, Independence Movement).
  • Conclusion: Because a single input produces multiple outputs, this relation is not a function.

• Topic-to-Course Relation (Function Example - Reversed Mapping):

  • Domain: Topics $\{\text{Ancient Civilization}, \text{Equations}, \text{Oral Traditions}, \text{Independence Movement}, \text{Graphs}\}$
  • Range: Courses $\{\text{African History}, \text{Algebra}\}$
  • Directional Mapping from Domain:
  • Ancient Civilization $\rightarrow$ African History
  • Equations $\rightarrow$ Algebra
  • Oral Traditions $\rightarrow$ African History
  • Independence Movement $\rightarrow$ African History
  • Graphs $\rightarrow$ Algebra
  • Evaluation:
  • "Ancient Civilization" $\rightarrow$ $1$ output ("African History")
  • "Equations" $\rightarrow$ $1$ output ("Algebra")
  • "Oral Traditions" $\rightarrow$ $1$ output ("African History")
  • "Independence Movement" $\rightarrow$ $1$ output ("African History")
  • "Graphs" $\rightarrow$ $1$ output ("Algebra")
  • Conclusion: Every single input in the domain maps to exactly one output in the range. Therefore, this reversed correspondence is a function. Multiple distinct inputs pointing to the same output course does not violate the definition of a function.

Analysis of Numerical and Conceptual Relations

• Numerical Mapping Example 1:

  • Domain: $\{8, 0, -5\}$
  • Range: $\{2\}$
  • Analysis: The input $8$ yields $2$; the input $0$ yields $2$; the input $-5$ yields $2$. Each individual domain element maps to exactly one output.
  • Conclusion: Is a function.

• Numerical Mapping Example 2:

  • Relation Condition: An input element (e.g., Jackson) has multiple outputs.
  • Analysis: A single domain value yields more than one output value.
  • Conclusion: Not a function.

• Set of Rectangles to Set of Areas Example:

  • Domain: A set of rectangles of varying dimensions (e.g., large rectangle, standard rectangle, tiny rectangle).
  • Range: A set of numerical area measurements (e.g., $24\,\text{inches}$, $10\,\text{inches}$, $20\,\text{inches}$).
  • Analysis: Any specific rectangle has exactly one calculated area. A single geometric rectangle cannot produce multiple distinct areas.
  • Conclusion: Is a function.

• Student Names to Birthdates Example:

  • Domain: Names of students in a class (e.g., Student $a$, Student $b$, Student $c$).
  • Range: Set of calendar dates in a given year.
  • Analysis: Each individual student has exactly one birthday. While two separate students can share the exact same birthday, no single student possesses multiple birthdays.
  • Conclusion: Is a function.

Ordered Pairs and Graphical Mapping

• Ordered Pair Format: Given as $(x, y)$, where $x$ is the domain element (input) and $y$ is the range element (output).
• Sample Set of Ordered Pairs:
  • Coordinates: $(1, 3)$, $(2, 3)$, $(3, 2)$, $(4, -2)$
  • Domain ($x$-values): $\{1, 2, 3, 4\}$
  • Range ($y$-values): $\{3, 2, -2\}$ (repeated values in range sets are not listed multiple times)
• Mapping Breakdown:
  • $1 \rightarrow 3$
  • $2 \rightarrow 3$
  • $3 \rightarrow 2$
  • $4 \rightarrow -2$
• Conclusion: Because each $x$-value maps to precisely one $y$-value, this relation is a function.

Function Notation and Graphical Evaluation

• Function Notation $f(x)$: Denotes the output value in the range corresponding to the input value $x$ in the domain.

• Reading Notation: Read as "$f$ of $x$" or "the value of $f$ at $x$".

• Important Distinction: The notation $f(x)$ does not mean $f \times x$ (it does not indicate multiplication).

• Relation to Equations: $f(x)$ represents the variable $y$ in standard Cartesian equations ($y = f(x)$).

• Graphical Evaluation of Outputs:

  • Finding $f(-1)$: Locate $x = -1$ on the x-axis and trace vertically to the graph. The corresponding y-value is $0$. Thus, $f(-1) = 0$
  • Finding $f(0)$: Locate $x = 0$ on the x-axis (y-intercept). The corresponding y-value is $3$. Thus, $f(0) = 3$
  • Finding $f(-3)$: Locate $x = -3$ on the x-axis and determine the matching y-value on the curve.
  • Finding $f(1)$: Locate $x = 1$ on the x-axis and determine the matching y-value on the curve.

• Reverse Evaluation (Finding Input $x$ given Output $y$):

  • Evaluation Task: Find $x$ such that $f(x) = -2$
  • Interpretation: Determine the value of $x$ when $y = -2$
  • Procedure:
  1. Locate $y = -2$ on the y-axis.
  2. Draw a horizontal reference line to intersect the graph curve.
  3. Trace vertically from the intersection point to the x-axis.
  4. Identify the x-coordinate of intersection.
  • Result: $x = 1.5$

Domain and Range of Continuous Graphs

• Continuous Graph Domain ($x$-axis span):

  • Covers all horizontal input values from the leftmost point of the graph to the rightmost point.
  • Boundary Limits: Extends from $x = -3$ to $x = 3$
  • Set-Builder Notation: $\{x \mid -3 \le x \le 3\}$ (read as "$x$ such that $x$ can take any value between $-3$ and positive $3$")

• Continuous Graph Range ($y$-axis span):

  • Covers all vertical output values from the lowest point of the graph to the highest point.
  • Boundary Limits: Extends from $y = -3$ to $y = 3$
  • Set-Builder Notation: $\{y \mid -3 \le y \le 3\}$ (read as "$y$ such that $y$ is between $-3$ and positive $3$")

• Endpoint Inclusion Rules for Notation:

  • Square Brackets [a, b]: Endpoint values $a$ and $b$ are included in the domain or range.
  • Parentheses (a, b): Endpoint values $a$ and $b$ are not included in the domain or range.

Algebraic Evaluation of Functions

• Function Definition: $f(x) = 3x - 5$ (or $y = 3x - 5$)
• Problem: Evaluate $f(7)$
• Step-by-Step Evaluation:
  1. Substitute $x = 7$ into the algebraic expression:      $f(7) = 3(7) - 5$
  2. Perform multiplication:      $f(7) = 21 - 5$
  3. Perform subtraction:      $f(7) = 16$
• Solution: $f(7) = 16$

Questions & Discussion

• Question: How should domain and range be written? Is interval notation using brackets acceptable alongside set-builder notation?

  • Response: Both set-builder notation and interval notation are valid mathematical formats. Square brackets indicate that endpoints are included in the interval, whereas parentheses indicate that endpoints are excluded. Always check the specific software formatting instructions (e.g., Pearson online homework system) to ensure compliance with the required input format.

• Question: Does $f(x)$ mean multiplication between $f$ and $x$?

  • Response: No. $f(x)$ is function notation signifying the value of the function $f$ evaluated at input $x$; it never represents multiplication.