Functions, Domain and Range, Intercepts, and Monotonicity

Interval Notation and Fundamental Definitions

Domain and Range Notation Requirements:
- All domains and ranges must be expressed in proper interval notation utilizing brackets or parentheses.
- Verbal descriptions such as "all real numbers" are mathematically non-standard for test and exam requirements and are strictly disallowed.
Types of Intervals:
- Closed Interval $[a, b]$ :
  - Represented graphically by two closed dots connected by a line.
  - Uses brackets $[$ and $]$ .
  - Includes both endpoints $a$ and $b$ in the interval, as well as every real number situated between them.
- Open Interval $(a, b)$ :
  - Represented graphically by open circles at the endpoints connected by a line.
  - Uses parentheses $($ and $)$ .
  - Excludes both endpoints $a$ and $b$ from the set.
- Mixed Interval:
  - Combines an open circle at one endpoint and a closed dot at the other (e.g., $(a, b]$ or $[a, b)$ ).
- Unbounded Interval:
  - Extends indefinitely toward positive infinity $\bgroup\infty\egroup$ or negative infinity $-\bgroup\infty\egroup$ .
  - Infinity is a conceptual representation of unboundedness rather than a real number; therefore, infinity and negative infinity must always be bounded by a parenthesis (e.g., $[a, \infty)$ or $(-\infty, b]$ ).

Algebraic Determination of Domain

Domain of Radical Functions (Square Roots):
- Given the function $f(x) = \sqrt{x - 5}$ :
- Squaring positive or negative numbers yields positive results (e.g., $2^2 = 4$ and $(-2)^2 = 4$ ).
- Taking the principal square root of a negative value is undefined within the real number system and produces imaginary numbers.
- To ensure real outputs, the expression under the square root (the radicand) must be non-negative: greater than or equal to zero ( $\ge 0$ ).
- Set up the inequality: $x - 5 \ge 0$
- Solve for $x$ by adding $5$ to both sides: $x \ge 5$
- Verification of inputs:
  - If $x = 5$ : $\sqrt{5 - 5} = \sqrt{0} = 0$
  - If $x = 6$ : $\sqrt{6 - 5} = \sqrt{1} = 1$
  - If $x = 10$ : $\sqrt{10 - 5} = \sqrt{5}$
- Domain expressed in interval notation: $[5, \infty)$ .
Domain of Rational Functions:
- Given the rational function $f(x) = \frac{x}{x + 4}$ :
- Division by zero is mathematically undefined, resulting in a value that does not exist (DNE).
- To find restricted values, set the denominator equal to zero: $x + 4 = 0$
- Subtract $4$ from both sides: $x = -4$
- At $x = -4$ , the evaluation $f(-4) = \frac{-4}{-4 + 4} = \frac{-4}{0}$ is undefined.
- The domain consists of all real numbers except $x = -4$
- Domain expressed in interval notation: $(-\infty, -4) \cup (-4, \infty)$ .
- The union symbol $\cup$ joins the interval strictly less than $-4$ with the interval strictly greater than $-4$ .

Real-World Domain Application: Geometry Constraints

Problem Context:
- A rectangular sheet of metal must be constructed such that its length is twice its width ( $l = 2w$ ).
- The sheet's total area must be strictly less than $72\,\text{sq ft}$ .
Constructing the Area Function:
- Standard area formula for a rectangle: $\text{Area} = l \times w$
- Substitute $l = 2w$ into the area formula: $\text{Area}(w) = (2w) \times w = 2w^2$
Deriving Domain Constraints for Width ( $w$ ):
- Set the area function strictly less than $72$ : $2w^2 < 72$
- Divide both sides by $2$ : $w^2 < 36$
- Take the square root of both sides: $w < 6$
- Physical dimensions cannot be negative; thus, the negative root $-6$ is omitted as non-physical.
- The lower boundary must be greater than zero ( $w > 0$ ), because a width of $0\,\text{ft}$ yields zero area, eliminating the physical sheet.
- Since the constraint states strictly less than $72\,\text{sq ft}$ , the value $6$ is excluded (uses a parenthesis).
- Domain of width $w$ in interval notation: $(0, 6)$ .

Domain, Range, and Intercepts from Graphs

Domain and Range from Graphical Representations:
- Domain: Read along the horizontal axis ( $x$ -axis) from left to right.
- Range: Read along the vertical axis ( $y$ -axis) from bottom to top.
- Example Graphic Analysis:
  - Graph endpoints on horizontal axis: Closed dot at $x = -3$ , open dot at $x = 2$ .
  - Graphical Domain: $[-3, 2)$ .
  - Graph vertical outputs: Lowest point at $y = -1$ (closed), highest point at $y = 4$ (closed).
  - Graphical Range: $[-1, 4]$ .
Properties of Intercepts:
- $x$ -intercept: Point where the graph intersects the $x$ -axis ( $y = 0$ ).
- $y$ -intercept: Point where the graph intersects the $y$ -axis ( $x = 0$ ).
- Uniqueness Rule: A graph can have multiple $x$ -intercepts (e.g., periodic sinusoidal functions), but at most one $y$ -intercept.
- If a graph had multiple $y$ -intercepts, it would fail the Vertical Line Test and cease to be a valid function.
- Format: Intercepts must always be written as explicit ordered pairs $(x, y)$ .
- Open Circle Exceptions: An open circle located directly on an axis does not count as an intercept because the function is not defined at that specific point.

Algebraic Determination of Intercepts

Given a linear relationship of the form $y = \frac{x}{2} - 18$ :
Calculating the $x$ -intercept:
- Set $y = 0$ : $0 = \frac{x}{2} - 18$
- Add $18$ to both sides: $18 = \frac{x}{2}$
- Multiply both sides by $2$ : $x = 36$
- Express as an ordered pair: $(36, 0)$ .
Calculating the $y$ -intercept:
- Set $x = 0$ : $y = \frac{0}{2} - 18$ $y = 0 - 18 = -18$
- Express as an ordered pair: $(0, -18)$ .

Monotonicity: Definitions of Increasing, Decreasing, and Constant Functions

Increasing Function:
- A function $f(x)$ is increasing on an open interval if for any two inputs $a$ and $b$ in the interval: $\text{If } a < b, \text{ then } f(a) < f(b)$
- As $x$ values move left-to-right (increase), output $y$ values move upward.
Decreasing Function:
- A function $f(x)$ is decreasing on an open interval if for any two inputs $a$ and $b$ in the interval: $\text{If } a < b, \text{ then } f(a) > f(b)$
- As $x$ values move left-to-right (increase), output $y$ values move downward.
Constant Function:
- A function $f(x)$ is constant on an open interval if for any two inputs $a$ and $b$ in the interval: $f(a) = f(b)$
- The slope is zero, creating a flat horizontal line segment where $y$ values do not change as $x$ increases.

Analyzing Graph Behavior: Increasing/Decreasing Intervals and Extrema

Interval Characterization:
- Sample Continuous Polynomial Graph Analysis:
  - Domain: $(-\infty, \infty)$
  - Range: $[-1, \infty)$
  - Increasing Intervals: $(1, 3) \cup (4, \infty)$
  - Decreasing Intervals: $(-\infty, 1) \cup (3, 4)$
- Parentheses Requirement: Intervals of increasing and decreasing behavior MUST always be written using open intervals (parentheses).
- At the exact transition points between increasing and decreasing states, the slope/rate of change is zero; therefore, the point itself is neither increasing nor decreasing and must be excluded.
Local Extrema:
- Local Minimum: Occurs where a function transitions from decreasing to increasing. It represents the lowest output value relative to adjacent points in its immediate neighborhood.
- Local Maximum: Occurs where a function transitions from increasing to decreasing. It represents the highest output value relative to adjacent points in its immediate neighborhood.
- Extrema: The collective term used to refer to both local maximums and local minimums.

Questions & Audience Discussion

Question: Why are parentheses always used instead of brackets when defining intervals where a function is increasing or decreasing?
- Response: At the exact vertex or turning point where a function changes direction from decreasing to increasing (or vice versa), the graph is instantaneously flat (slope equal to zero). Because it is not actively rising or falling at that single point, the turning point cannot be classified as increasing or decreasing and must be excluded using an open parenthesis.
Question: What happens if a graph has an open circle at an axis intersection?
- Response: An open circle indicates that the function is undefined at that specific point. Consequently, if an open circle falls on the $x$ or $y$ axis, it does not constitute a valid intercept.
Question: Does a local minimum have to be the lowest point on the entire graph?
- Response: No. A local minimum is relative only to its immediate open interval section of the graph. A graph can have multiple local minimums, while the global or absolute minimum refers strictly to the lowest point across the entire domain.