Functions and Symmetry — Comprehensive Lecture Notes

Accessing Materials and Quick Review

• There are access issues with the Canvas module OneNote link: when opened, OneNote can be picky about view permissions based on email settings. If you can access it, it works; otherwise you might see clipping or missing content. This is a practical reminder to verify access before class reminders or submissions.
• The class largely reviewed basic notation for functions and the idea that a function maps each x-value to exactly one y-value. This is the cornerstone of the vertical line test and function notation.

Core Concept: What is a Function?

• Definition in words: Every x-value has exactly one corresponding y-value.
• Graphically: If you imagine drawing vertical lines through the graph, each vertical line should intersect the graph at exactly one point.
• Consequence: If a vertical line hits the graph in more than one place, the relation is not a function.
• Related idea: A function is a special kind of relation where the mapping from x to y is single-valued for every x.

Vertical Line Test (VLT)

• Method: For any x-value, draw (or imagine) a vertical line through that x, and see how many times it meets the graph.
• If every vertical line intersects the graph exactly once, the relation is a function.
• If any vertical line intersects more than once, the relation is not a function.
• Note: This test applies whether you have a picture, an equation, or a list of points.

Reading a Graph: Notation and What to Find

• Given a graph, you may be asked to identify several quantities. For the example shown in class, these were labeled parts a–f.
• The key quantities to read from a graph include:
  • f(−5), f(0), f(3): the y-values corresponding to specific x-values (read off the graph).
  • Domain: all x-values that actually appear on the graph (i.e., the set of x for which there is a point on the graph).
  • Range: all y-values that occur on the graph.
  • X-intercepts: points where the graph crosses the x-axis (y = 0).
  • Y-intercepts: points where the graph crosses the y-axis (x = 0).
  • Intersections with a horizontal line y = c (i.e., how many times the graph crosses a line y = c).
• For a graph with a marked x-value (an x-value between known axis marks): follow that x value upward to meet the curve to find the corresponding y-value.

Worked Example from the Lecture (Using a Specific Graph)

• Given a graph where the right-most labeled y-value on the top reaches 8 (the highest y), and the graph has a minimum y-value below zero:
  • f(−5) = 8. This is read by locating x = −5 on the x-axis and tracing up to the graph: y = 8.
  • f(0) and f(3) are read the same way from the graph; the domain and range depend on the graph's extent.
• Domain and Range (from the described graph):
  • Domain: x-values that appear on the graph are from x = −7 up to x = 3 (inclusive). This can be written as:
  • Interval form: [-7,3]
  • Set-builder form: { x \mid -7 \le x \le 3 }
  • Range: y-values that appear on the graph include the maximum of 8 and a minimum value that lies below zero (the exact minimum is the lowest y-value shown on the graph). Thus:
  • Range is from the minimum y-value to the maximum y = 8, i.e., [y{\min}, 8] where y{\min} < 0 and is the graph’s lowest point.
• X-intercepts (where f(x) = 0): the points where the graph crosses the x-axis. In this example, the x-intercepts are at x = 0 and x = −1, giving intercept points (-1,0) and (0,0).
• Y-intercept (where x = 0): read off the graph if needed. In many graphs, you’ll also note this value as the y-intercept, but it’s not always present as a distinct labeled point if the axis crossing doesn’t occur on the curve.
• Intersections with horizontal line y = 2: count how many times the graph crosses this line. In the described case, the line y = 2 intersects the graph at three points (three x-values).
• Reading the line y = 2 and finding x values where f(x) = 2:
  • Solve the equation f(x) = 2.
  • In the worked example, this involved clearing denominators: if f(x) = \frac{x+1}{x+2}, then
    \frac{x+1}{x+2} = 2 \Rightarrow 2(x+2) = x+1 \Rightarrow 2x+4 = x+1 \Rightarrow x = -3.
• Reading the line y = 0 (x-intercepts) again: solve \frac{x+1}{x+2} = 0 \Rightarrow x+1 = 0 \Rightarrow x = -1. The corresponding x-intercept is (-1,0).
• Summary of the steps used in this activity:
  • Identify points via reading the graph.
  • Determine domain and range from the graph’s extent.
  • Use the definition f(a) = y to evaluate specific x-values.
  • Solve equations by substituting into the function (or by algebraic manipulation if given as a fraction).
  • Distinguish between x-intercepts (where y = 0) and zeros (the x-values that make f(x) = 0).

Notation Choices and Practice with MyMathLab

• Two common ways to present domain information:
  • Interval notation: (-\infty, a) \cup (b, \infty) (for a denominator restriction like x ≠ a, you remove a from the real line).
  • Set-builder notation: { x \mid \text{condition} }, e.g., { x \mid x \neq -2 } for the domain of a rational function with a pole at x = −2.
• When you are asked to write the domain, some courses want interval notation, some want set-builder notation; both are acceptable if the assignment specifies.
• Example domain for a function with a nonzero denominator except at x = −2:
  • Interval notation: (-\infty, -2) \cup (-2, \infty)
  • Set-builder: { x \in \mathbb{R} \mid x \neq -2 }
• Exercises often require you to:
  • Compute f(1) and verify whether a given point lies on the graph by checking if the computed value equals the y-coordinate of the point.
  • Check the point (1, 1/2) for membership on the graph of f(x) = \frac{x+1}{x+2} by evaluating f(1) = \frac{2}{3} and noting that 1/2 ≠ 2/3, so the point is not on the graph.
• In class, the instructor emphasized flexibility in answer formats on MyMathLab: you may be asked to write in set-builder form or interval form depending on the prompt. Both are acceptable if the prompt allows.

Introduction to Section 1.3: Different Properties of Functions

• This section moves from basic definitions to exploring symmetry in functions.
• The main goal: understand symmetry as a property of the graph and how to recognize it from the equation or the graph.

Symmetry of Functions: Key Concepts

• Symmetry with respect to the x-axis
  • Definition (graphical): If you replace y with −y in the equation, you obtain an equivalent equation. In geometric terms, the graph is mirror-symmetric across the x-axis (top and bottom halves match).
  • Visual example: A point (x, y) on the graph has a symmetric point (x, −y).
  • Examples: If a point on the graph is (1, 1), a symmetric point across the x-axis is (1, −1). If a point is (4, 2), its symmetric point is (4, −2).
• Symmetry with respect to the y-axis
  • Definition: If you replace x with −x in the equation, you obtain an equivalent equation. The graph is mirror-symmetric across the y-axis (left and right halves match).
  • Visual example: A point (x, y) has symmetric point (−x, y).
  • Examples: (1, 1) ↔ (−1, 1); (4, 2) ↔ (−4, 2).
• Symmetry with respect to the origin
  • Definition: If you replace x with −x and y with −y, you obtain an equivalent equation. The graph is symmetric through the origin (rotate 180 degrees about the origin).
  • Visual example: A point (x, y) has symmetric point (−x, −y).
  • Example: (2, 8) ↔ (−2, −8).
• A graph can have none, one, or all three symmetry types. A circle, for example, has all three symmetries; some curves have only one or none.
• Practical takeaway: Symmetry properties are often tested first by looking at the equation (algebraic check) and then by visual inspection of the graph. Algebraic testing involves replacing x with −x (for y-axis), replacing y with −y (for x-axis), or replacing both (for origin).

Applying Symmetry: Quick Visuals from the Lecture

• If a graph is symmetric about the x-axis, folding the graph along the x-axis maps the top to the bottom.
• If symmetric about the y-axis, folding along the y-axis maps the left to the right.
• If symmetric about the origin, a 180-degree rotation about the origin maps the graph onto itself.
• A graph that is symmetric about all three axes (x, y, and origin) often resembles a circle or a highly symmetric shape like a clock-face, whereas many common functions (like polynomials of odd degree, etc.) have limited or no symmetry.

Quick Tips for Class Practice and Homework

• When you’re unsure about notation, ask neighbors or the instructor to clarify parts (a)–(f) in the graph-reading tasks.
• For x-intercepts and zeros, remember: x-intercepts are the x-values where y = 0; zeros are the values of x that satisfy f(x) = 0. They are often the same numerical value, but sometimes terminology varies (x-intercept as a point vs. zero as the x-value).
• When solving fractional equations (like f(x) = 2 with f(x) = \frac{x+1}{x+2}), clear the fractions by multiplying both sides by the common denominator to avoid mistakes.
• In class, you’ll use both interval notation and set-builder notation for domain and range; know how to convert between them:
  • Interval: (-\infty, -2) \cup (-2, \infty)
  • Set-builder: { x \in \mathbb{R} \mid x \neq -2 }
• For MyMathLab submissions, you may be asked to provide a single concise answer or show the steps; adapt to the prompt while keeping the mathematical reasoning clear.

Summary of Key Formulas and Concepts

• Function definition (single-valued mapping):

  • For every x, there exists a unique y with y = f(x).

• Vertical Line Test (VLT):

  • A graph is a function iff every vertical line intersects it at most once.

• Domain and Range from a graph:

  • Domain: { x \mid x \text{ is used by the graph} } or interval form [a, b] ranges depending on the picture.
  • Range: set of all y-values attained by the graph.

• Examples (from the worked graph):

  • Read f(−5) = 8: f(-5)=8
  • Domain: [-7, 3]; Range: [y{\min}, 8] with y{\min} < 0 (lowest y-value on the graph).
  • X-intercepts: (-1,0) and (0,0) .
  • Evaluate f(1): if f(x)=\dfrac{x+1}{x+2}, then f(1)=\dfrac{2}{3}, so the point $(1,\tfrac{1}{2})$ is not on the graph.
  • If f(2)=\dfrac{3}{4}, then the point $(2, \tfrac{3}{4})$ is on the graph.
  • Solve \dfrac{x+1}{x+2} = 2: 2(x+2) = x+1 \Rightarrow x = -3; hence the point where f(x) = 2 is x = -3.
  • Solve for zeros: \dfrac{x+1}{x+2} = 0 \Rightarrow x = -1.

• Symmetry quick map:

  • x-axis symmetry: replace y by −y; (x, y) ↔ (x, −y) for points on the graph.
  • y-axis symmetry: replace x by −x; (x, y) ↔ (−x, y).
  • Origin symmetry: replace x by −x and y by −y; (x, y) ↔ (−x, −y).

• Common classroom tweaks:

  • Some courses prefer set-builder notation for domain/range; others prefer interval notation. Both are acceptable when specified.
  • When in doubt about notations, use the instructor’s preferred form or include both forms if allowed.

If you want, I can tailor these notes to focus more on a specific problem set or convert more of the lecture’s examples into explicit LaTeX-ready steps and answers for quick study review.