Comprehensive Notes on Function Symmetry, Behavior, Extrema, Piecewise Functions, and Function Combinations

Algebraic Determination of Function Symmetry

• Challenges of Determining Symmetry from Equations:

  • Visual inspection of a graph readily reveals symmetry, but equations do not show the coordinates directly, requiring algebraic analysis or plotting specific test points on a Cartesian plane.
  • Graphing every coordinate of an equation is inefficient; selecting targeted coordinates ($x$ and $-x$) allows quick determination of symmetry.

• Fundamental Definitions of Symmetry:

  • Even Symmetry ($y$-axis symmetry): A relation or function is even if every point $(x, y)$ corresponds to the point $(-x, y)$. When graphed, the coordinates reflect identically across the vertical $y$-axis.
  • Odd Symmetry (Origin symmetry): A relation or function is odd if every point $(x, y)$ corresponds to the point $(-x, -y)$. The coordinates reflect across the origin, producing opposite signs for both coordinates.
  • $x$-Axis Symmetry: A relation exhibits $x$-axis symmetry if every point $(x, y)$ corresponds to $(x, -y)$. A non-trivial graph with $x$-axis symmetry fails the vertical line test because a single $x$-value pairs with two distinct $y$-values, meaning it is not a function. In problems restricted to functions, this outcome indicates the relation is neither even nor odd.

• Symmetry Testing Rules for Functions vs. Non-Functions:

  • If an equation is explicitly defined as a function, establishing one symmetry type allows testing to stop immediately, as a valid function cannot possess multiple symmetries (such as both $y$-axis and $x$-axis symmetry).
  • If an equation is not designated as a function, it can possess all three symmetries simultaneously, requiring all symmetry definitions to be evaluated before concluding.

• Selecting Test Values:

  • The recommended standard test values are $x = 1$ and $x = -1$.
  • The value $x = 0$ must be avoided because zero is neutral (neither positive nor negative) and lies directly on the axis of symmetry, obscuring directional reflections across the vertical axis.

• Step-by-Step Algebraic Evaluation of an Example Equation:

  • Given equation: $y = 6x^6 + 3x^2 + |x|$
  • Order of Operations: Follow standard algebraic order of operations (PEMDAS), calculating exponents before multiplication. The absolute value function represents the direct distance from zero on a number line.
  • Evaluating at $x = 1$:
    • Evaluate exponents: $1^6 = 1$ and $1^2 = 1$.
    • Evaluate absolute value: $|1| = 1$.
    • Substitute and multiply: $y = 6(1) + 3(1) + 1$.
    • Compute the sum: $y = 6 + 3 + 1 = 10$.
    • Resulting coordinate: $(1, 10)$.
  • Evaluating at $x = -1$:
    • Evaluate exponents: $(-1)^6 = 1$ and $(-1)^2 = 1$ (raising a negative number to an even power yields a positive value).
    • Evaluate absolute value: $|-1| = 1$.
    • Substitute and multiply: $y = 6(1) + 3(1) + 1$.
    • Compute the sum: $y = 6 + 3 + 1 = 10$.
    • Resulting coordinate: $(-1, 10)$.
  • Conclusion: Since $x = 1$ and $x = -1$ yield the identical output $y = 10$, the coordinates $(1, 10)$ and $(-1, 10)$ confirm that the equation is an even function with $y$-axis symmetry.

Intervals of Increase, Decrease, and Constant Behavior

• Foundational Directional Rule:

  • Graphs must strictly be read and analyzed from left to right (in order of increasing values of $x$).
  • Reading graphs from bottom-to-top causes curves to falsely appear permanently increasing.
  • Reading graphs from right-to-left reverses the mathematical definitions and yields entirely incorrect intervals.

• Definitions of Function Behavior:

  • Increasing Function: The output values ($y$) continuously rise as the input moves from left to right.
  • Decreasing Function: The output values ($y$) continuously fall as the input moves from left to right.
  • Constant Function: The output values ($y$) remain identical (horizontal line) as the input moves from left to right.

• Formatting Intervals of Function Behavior:

  • Input vs. Output: Always analyze the behavior of $y$, but record the intervals exclusively using the corresponding $x$-values.
  • Parentheses vs. Brackets: Intervals of increase, decrease, and constant behavior must always use parentheses ( ) and never brackets [ ].
  • Mathematical Justification for Parentheses: At the exact endpoint where an interval terminates, the function is transitioning or switching behaviors. At that instantaneous point, the function is neither increasing, decreasing, nor constant; it serves solely as a boundary.
  • Infinities: Positive infinity ($\infty$) and negative infinity ($-\infty$) always take parentheses because they represent boundless concepts rather than discrete numeric values.
  • Combining Disjoint Intervals: When a function exhibits a specific behavior across multiple separate sections, intervals are separated using a comma (e.g., $(-\infty, a), (b, \infty)$) or a union symbol ($\cup$).

• Comprehensive Graph Analysis Example:

  • Left Section: An arrow extends upward to the left. Tracing left to right, the outputs fall through $y = 5, 4, 3, 2$ down to $y = 0$ at the input $x = 0$. This segment is decreasing on the interval $(-\infty, 0)$.
  • Middle Section: Tracing left to right from $x = 0$ to $x = 2$, the outputs remain stationary at $y = 0$. This segment is constant on the interval $(0, 2)$.
  • Right Section: Tracing left to right from $x = 2$ onwards, the outputs drop through $y = 0, -1, -2, -3, -4$ toward negative infinity ($-\infty$). This segment is decreasing on the interval $(2, \infty)$.
  • Final Categorization:
    • Increasing: Never (Interval does not exist).
    • Decreasing: Sometimes decreasing on $(-\infty, 0), (2, \infty)$.
    • Constant: Sometimes constant on $(0, 2)$.

Local Extrema: Relative Maxima and Minima

• Distinction Between Relative (Local) and Absolute (Global) Extrema:

  • Relative extrema do not require finding the overall absolute highest or lowest points across the entire domain.
  • A relative extremum represents a local high or low point within an immediate neighborhood of the graph.

• Definitions and Characteristics:

  • Relative Maximum:
    • A local peak or high point on a graph.
    • Occurs precisely where a function transitions from increasing to decreasing.
    • Multiple relative maxima can exist within a single function.
  • Relative Minimum:
    • A local valley or low point on a graph.
    • Occurs precisely where a function transitions from decreasing to increasing.
    • Multiple relative minima can exist within a single function.

• Mathematical Terminology Specifications:

  • Location: Refers specifically to the independent input variable, $x$.
  • Value: Refers specifically to the dependent output variable, $y$.
  • Point: Refers to the complete ordered pair, $(x, y)$.
  • Singular vs. Plural: "Maximum" and "minimum" denote singular occurrences, whereas "maxima" and "minima" denote plural occurrences.

• Detailed Graph Analysis Example (Example 4):

  • First Transition: From $x = -\infty$ to $x = -2$, the graph decreases. From $x = -2$ to $x = 0$, the graph increases. The transition from decreasing to increasing forms a valley at ordered pair $(-2, 0)$. This is a relative minimum with location $x = -2$ and value $y = 0$.
  • Second Transition: From $x = -2$ to $x = 0$, the graph increases. From $x = 0$ to $x = 2$, the graph decreases. The transition from increasing to decreasing forms a peak at ordered pair $(0, 2)$. This is a relative maximum with location $x = 0$ and value $y = 2$.
  • Third Transition: From $x = 0$ to $x = 2$, the graph decreases. Beyond $x = 2$, the graph increases toward $\infty$. The transition from decreasing to increasing forms a valley at ordered pair $(2, 0)$. This is a second relative minimum with location $x = 2$ and value $y = 0$.
  • Summary of Extrema:
    • The function possesses two relative minima located at the points $(-2, 0)$ and $(2, 0)$.
    • The function possesses one relative maximum located at the point $(0, 2)$.

Piecewise Functions: Principles, Real-World Applications, and Graphing Procedures

• Definition and Domain Constraints:

  • A piecewise function is a single function composed of multiple sub-functions, where each sub-function applies to a distinct, non-overlapping interval of the domain.
  • Necessity of Unique Domains: Sub-functions cannot share overlapping domain values; an overlapping $x$-value would yield multiple distinct $y$-values, violating the definition of a function.

• Real-World Applications:

  • Overtime Pay: Hourly earnings represent a piecewise function where hours worked $x \le 40$ are calculated at a standard base hourly rate, while hours worked $x > 40$ are calculated at an overtime rate.
  • Water Utility Bills: Usage is billed at stepped consumption tiers where rates shift once usage crosses specific volumetric thresholds.
  • Progressive Income Tax: The tax system utilizes distinct marginal tax brackets (currently seven separate income brackets), with each tier functioning as an individual sub-function over a defined income domain.

• Step-by-Step Graphing of a Piecewise Function (Example 5):

  • Given piecewise definition:         $f(x) = \begin{cases} -x + 3 & \text{if } x \le 3 \\ \sqrt{x - 3} & \text{if } x > 3 \end{cases}$

  • Sub-Function 1 (Linear Ray): $y = -x + 3$ on domain $(-\infty, 3]$

    • The domain boundary at $x = 3$ includes an equality ($\le$), dictating a solid, closed circle endpoint.
    • Construct a table of values moving left to right up to the boundary $x = 3$:
      • Let $x = -2$: $y = -(-2) + 3 = 2 + 3 = 5 \implies (-2, 5)$
      • Let $x = 0$: $y = -(0) + 3 = 3 \implies (0, 3)$
      • Let $x = 3$: $y = -(3) + 3 = 0 \implies (3, 0)$
    • Plot these points and connect them as a continuous ray starting at the closed endpoint $(3, 0)$ and extending infinitely upward and to the left through $(0, 3)$ and $(-2, 5)$.

  • Sub-Function 2 (Radical Curve): $y = \sqrt{x - 3}$ on domain $(3, \infty)$

    • The domain boundary at $x = 3$ is a strict inequality ($>$), indicating an open circle endpoint.
    • Construct a table of values starting at $x = 3$ and increasing to the right:
      • Let $x = 3$: $y = \sqrt{3 - 3} = \sqrt{0} = 0 \implies (3, 0)$
      • Open vs. Closed Circle Intersection: The open circle at $(3, 0)$ from the second piece coincides precisely with the closed circle at $(3, 0)$ from the first piece. The closed circle fills the open circle, creating a continuous junction.
      • Let $x = 4$: $y = \sqrt{4 - 3} = \sqrt{1} = 1 \implies (4, 1)$
      • Let $x = 5$: $y = \sqrt{5 - 3} = \sqrt{2} \approx 1.414$ (irrational decimal; skipped to prioritize whole numbers for precise plotting).
      • Let $x = 6$: $y = \sqrt{6 - 3} = \sqrt{3} \approx 1.732$ (irrational decimal; skipped).
      • Let $x = 7$: $y = \sqrt{7 - 3} = \sqrt{4} = 2 \implies (7, 2)$
    • Plot $(4, 1)$ and $(7, 2)$, connecting them to $(3, 0)$ to form the characteristic half-curve of a square root function extending toward positive infinity.

• Digital Graphing Tools and Execution:

  • Ray Tool: Used for linear functions defined over a semi-infinite interval; requires placing an initial endpoint and clicking a secondary point along the path to project the ray.
  • Square Root Tool: Plots a radical curve from a defined initial coordinate.
  • Scissor/Cut Tool: Utilized when graphing piecewise components involving parabolas ($x^2$). Standard parabola tools plot complete $U$-shaped curves; the scissor button cuts away the unused half of the curve that lies outside the specified piecewise domain.

Classroom Practice and Evaluation Checks

• Poll Everywhere Session Details:

  • Student responses submitted via access portal pollev.com/andyvong59eight.

• Conceptual Check 1: Definition of a Piecewise Function:

  • Correct Formulation: A function that utilizes different formulas, rules, or equations for distinct parts or intervals of its domain.

• Visual Check 2: Identifying Piecewise Function Graphs:

  • Correct Selection: Identifying graphs divided into distinct segments or multiple distinct sub-function curves.
  • Common Parent Functions (Incorrect Distractors): Continuous single-rule functions such as the linear identity function ($y = x$) and the cubic parent function ($y = x^3$) do not constitute piecewise graphs.

• Computational Check 3: Evaluating a Specific Domain Value:

  • Problem Prompt: Evaluate a three-part piecewise function at $x = 4$.
  • Domain 1: $x < -6$. Test: Is $4 < -6$? No. This sub-function cannot be used.
  • Domain 2: $-6 < x < 6$ (or $-6 \le x < 6$). Test: Is $-6 < 4$ and $4 < 6$? Yes. The input $x = 4$ lies within this interval.
  • Sub-Function Formula for Domain 2: $y = -9x + 2$
  • Calculation:$y = -9(4) + 2$$y = -36 + 2 = -34$
  • Performance Metric: $67\%$ of the class correctly identified the appropriate domain piece and computed the result $-34$.

Introduction to Section 2.8: Algebra of Functions and Function Composition

• Course Scope and Exam Timeline:

  • Lecture recorded on 10/20/2026.
  • Section 2.8 is the final topic of Chapter 2.
  • Second exam scheduled for Tuesday, 10/27/2026. Coverage includes all course material completed through the lecture on Thursday, 10/22/2026 (anticipated completion of Section 2.8 and commencement of Section 3.1; material up to 4.1 not covered will roll over to Exam 3).

• Two Core Mathematical Operations on Functions:

  1. Algebra of Functions (Combination Functions):

     • Functions constructed via the four elementary operations of arithmetic: addition, subtraction, multiplication, and division ($+, -, \times, \div$).
     • Exponents and Radicals: Raising an expression to a power represents repeated multiplication, and taking an $n$-th root seeks factors that multiply to produce the radicand; both operations are extensions of algebraic multiplication.
     • Domain Rule: The domain of an algebraic combination of functions consists solely of the set of all real numbers $x$ where the domains of both individual functions overlap (the intersection of their domains):             $\text{Domain} = \text{Domain}(f) \cap \text{Domain}(g)$(For division $\frac{f}{g}$, values where the denominator $g(x) = 0$ must also be excluded).

  2. Function Composition:

     • Functions constructed not by arithmetic operations, but through substitution, wherein an entire function replaces the independent variable $x$ within another function ("a function within a function").
     • Musical Composition Analogy: Parallels composing music, where multiple distinct melodies, rhythms, and sounds are integrated together to form a unified work.
     • Domain Rule: Determining the domain of a composite function $(f \circ g)(x) = f(g(x))$ requires a two-step restriction:
       1. Determine the domain of the simplified composite expression itself.
       2. Subtract (exclude) any values of $x$ that are not contained within the domain of the internal replacement function ($g(x)$).             Core Axiom: An input value cannot exist in the final composite function if it is fundamentally undefined at the point of initial substitution into the interior function:             $\text{Domain}(f \circ g) = \{x \in \text{Domain}(g) \mid g(x) \in \text{Domain}(f)\}$