Exhaustive University Study Notes on Domain and Range

Core Definitions of Domain and Range

Domain:
- The set of all valid inputs ( $x$ -values) that are involved in or accepted by a function.
- Represents all the values placed into an equation or function.
Range:
- The set of all valid outputs ( $y$ -values) produced by a function.
- Represents all the resulting values that come out of an equation or function.

Graph Compression Method for Determining Domain and Range

Determining Domain Visually ("Horizontal Compression Method"):
- Take the graph of the function and conceptually squish, collapse, or compress it vertically onto the horizontal $x$ -axis.
- Identify which sections of the $x$ -axis are physically covered by the compressed graph.
- The exact $x$ -values covered during this compression constitute the domain of the function.
- Any region along the $x$ -axis where no graph material compresses down is excluded from the domain.
Determining Range Visually ("Vertical Compression Method"):
- Take the graph of the function and conceptually smoosh, collapse, or compress it horizontally onto the vertical $y$ -axis.
- Identify which sections of the $y$ -axis are covered by the compressed graph.
- The exact $y$ -values covered during this compression constitute the range of the function.
- Overlap is permissible in range determination: if multiple distinct segments of a graph collapse onto the exact same section of the $y$ -axis, that region is included in the range without needing to be listed multiple times.

Rules for Interval Notation and Boundary Points

Directional Ordering:
- Domain ( $x$ -axis): Write intervals from left to right (from smaller numbers to larger numbers, moving from $-\text{infinity}$ / $-\frac{\text{left}}{\text{side}}$ toward $\text{positive infinity}$ / $\text{right side}$ ).
- Range ( $y$ -axis): Write intervals from bottom to top (from smaller numbers to larger numbers, moving from negative numbers at the bottom up to positive numbers at the top / $\text{positive infinity}$ ).
Boundary Circle Types:
- Closed Circle (Filled Dot): Indicates that the endpoint value is included in the domain or range. Represented in interval notation using square brackets [ or ].
- Open Circle (Unfilled Hole): Indicates that the endpoint value is excluded from the domain or range. Represented in interval notation using parentheses ( or ).
- Infinities ( $-\text{infinity}$ and $\text{positive infinity}$ ): Infinities always take parentheses ( or ) because infinity is an unbounded direction rather than a fixed number.
Continuous Graph Lines:
- A solid, unbroken line on a graph represents an infinite series of adjacent closed points.
- Unless explicitly interrupted by an unfilled hole or open circle, extrema (peaks and troughs) or segments along a solid line default to included closed boundaries ([ or ]).

Overlap, Discontinuities, and Interval Simplification Rules

The "Closed Circle Wins" Rule:
- When an open circle and a closed circle occur at the exact same location (same $x$ -value or $y$ -value) across different pieces of a graph, the closed circle fills in the open circle.
- Because the value is included via the closed circle, coverage across that point is continuous, and the point is fully included in the interval.
Simplification of Overlapping Intervals:
- When collapsing graph segments produces overlapping intervals, combine them into the single largest continuous interval possible.
- Even if a section of an axis is covered multiple times by different pieces of the graph, it is written as one simplified interval.
Handling Breaks, Holes, and Gaps:
- Any physical gap, hole, or break in the graph that leaves a portion of an axis uncovered requires breaking the notation into separate distinct intervals.
- Combine non-contiguous valid intervals using the union symbol $\bigcup$ .

Comprehensive Worked Practice Examples

Example 1: Discontinuous Domain with Shared Range
- Domain Analysis:
- Left piece extends endlessly to the left from $x = -1$ (marked with a closed circle), covering $(-\text{infinity}, -1]$ .
- A gap exists between $x = -1$ and $x = 1$ where no graph exists.
- Right piece starts at $x = 1$ (marked with a closed circle) and extends endlessly to the right, covering $[1, \text{infinity})$ .
- Combined Domain: $(-\text{infinity}, -1] \bigcup [1, \text{infinity})$ .
- Range Analysis:
- Collapsing both pieces horizontally onto the vertical axis shows that both segments start at $y = 0$ (closed circle) and extend upward endlessly toward positive infinity.
- No $y$ -values below $0$ are covered.
- Simplified Range: $[0, \text{infinity})$ .
Example 2: Overlapping Pieces and Continuous Domain
- Domain Analysis:
- First piece compresses to cover the left portion of the $x$ -axis up to a boundary point.
- Second piece compresses to cover the right portion starting at the same $x$ -value.
- One segment ends with a closed circle while the other segment starts with an open circle at the same $x$ -value; by the "Closed Circle Wins" rule, the point is filled in.
- Entire axis is covered without interruption.
- Simplified Domain: $(-\text{infinity}, \text{infinity})$ .
- Range Analysis:
- First piece covers $[1, \text{infinity})$ .
- Second piece starts at an open circle at $y = -2$ and covers upwards through $(-2, \text{infinity})$ .
- The section from $y = 1$ to $\text{infinity}$ is covered twice, but overall coverage spans continuously from $y = -2$ to $\text{infinity}$ .
- Simplified Range: $(-2, \text{infinity})$ .
Example 3: Finite Piecewise Graph with Vertical Gap
- Domain Analysis:
- Graph starts at $x = -3$ (closed circle).
- Continuous curve fills in all intermediate values and terminates at $x = 2$ (closed circle).
- Combined Domain: $[-3, 2]$ .
- Range Analysis:
- Lower segment spans from $y = -3$ (closed circle) up to $y = 2$ (parenthesis/open boundary), giving $[-3, 2)$ .
- A gap exists on the vertical axis, then coverage resumes at $y = -1$ (closed boundary at the trough of the curve) and extends up to a peak at $y = 3$ (closed continuous curve peak).
- Range Intervals: $[-3, 2)$ and $[-1, 3]$ .
Example 4: Continuous Curve with Open Endpoint
- Domain Analysis:
- Starts at an open circle at $x = -2$ and covers all values up to x = 4$.\n - Domain: (-2, 4).\n - **Range Analysis**:\n - Smallest y $-value occurs at the bottom trough of the graph at$ y = -1 (solid continuous line, closed boundary).\n - Graph extends upward to a maximum height at y = 15, terminating at an open circle.\n - Range: [-1, 15).\n\n- **Example 5: Domain Discontinuity via Aligned Open Circles**\n - **Domain Analysis**:\n - Graph begins at x = -3 and moves rightward.\n - Encounters a point where both upper and lower graph segments have open circles at the exact same x-value, creating an un-filled hole/break in the domain.\n - The break forces the domain to be split into two separate intervals joined by a union symbol igcup\n\n\n# Questions & Discussion\n\n- **Question**: When a graph contains an open circle hole, how do we determine where closed circles start and end for continuous portions of the graph?\n- **Answer**: If a point represents the bottom, top, or solid line of a graph curve, it is included as closed (`[` or `]`) because a continuous line consists of infinitely many adjacent closed points. Even if an endpoint elsewhere on the graph has an open circle, any solid section of the curve that hits a minimum or maximum y $-value is considered closed at that extremum (for instance, a trough at$ y = -1 is closed because it is part of the solid line of the graph).\n\n\n# Logistics and Examination Details\n\n- **Quiz 1 Schedule**: Quiz 1 takes place on Friday.\n- **Quiz Format**:\n - Short quiz consisting of exactly 2$$ questions.
Practice Problem Sheet:
- The practice problem sheet has more parts and questions than the actual quiz will contain to provide extra practice.
- Students are instructed to complete practice problems independently prior to reviewing published solutions.
- Answer solutions for Practice Quiz 1 will be posted on Canvas in the near future.