Comprehensive Notes: Section 1.1–1.3 — Functions, Graphs, and Course Logistics

Section 1.1–1.3: Functions, Graphs, and Core Properties (Comprehensive Notes)

  • Overview: Transcript covers introduction to relations and functions, interval notation, operations on functions, the graph of a function, domain/range concepts from graphs, and properties of functions (even/odd, symmetry, monotonicity, local/absolute extrema, and average rate of change). It also includes practical course logistics for an online course using Webex, iCollege, and Pearson/MyLab Math, plus attendance and access steps.

1.1: Relations and Functions

  • Relation definition

    • A relation is a correspondence between two sets: Domain (inputs) X and Range (outputs) Y.
    • Written as ordered pairs (x, y) with the arrow x → y indicating x relates to y.
    • In a relation, each x in the domain relates to at least one y in the range.
    • Notation: x is the input (domain), y is the output (range).

  • Function definition

    • A function is a relation with the extra condition: each element of the domain relates to exactly one element of the range.
    • Key distinction: a function cannot assign two outputs to the same input.
    • Example intuition: vending machine analogy. A function maps each code (input) to exactly one snack (output). Different codes can map to the same snack, but a given code cannot map to two snacks.

  • Function notation

    • Use f, g, h to denote functions.
    • For a value x in the domain, the corresponding output is written as f(x). This is read as “f of x.”
    • Notation note: f(x) is not multiplication; it is the function value at input x.

  • Simple example

    • If f(x) = x^2 + 1, then the function value at x = 4 is:

    f(4)=42+1=17.f(4) = 4^2 + 1 = 17.

  • Composite input example

    • If the same function is used and you want f(3x), you substitute the input with 3x:

    • For f(u) = u^2 + 1, we get:

    f(3x)=(3x)2+1=9x2+1.f(3x) = (3x)^2 + 1 = 9x^2 + 1.

  • Difference Quotient (a core concept in Section 1.1)

    • Purpose: measure the average rate of change of a function over a small input change h.
    • Definition:

    rac{f(x+h) - f(x)}{h}

    • Example with f(x) = x^2 + 1:
    • f(x+h) = (x+h)^2 + 1 = x^2 + 2xh + h^2 + 1.
    • f(x) = x^2 + 1.
    • Difference: f(x+h) - f(x) = (2xh + h^2).
    • Divide by h:

    f(x+h)−f(x)h=2xh+h2h=2x+h.\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2}{h} = 2x + h.

  • Domain considerations (brief reminder)

    • When finding the domain, exclude inputs that cause division by zero or invalid operations (e.g., even roots of negative numbers).

  • Quick reminder on intervals and limits (to connect with later sections)

    • Interval notation is used to describe sets of real numbers in a compact form (see Section 1.1 on Interval Notation).

Interval Notation (brief recap from 1.1)

  • Real line view: from smaller to larger values.
  • Notation basics:
    • Endpoints included: use brackets [a, b].
    • Endpoints not included: use parentheses (a, b).
    • Infinity symbols are always written with parentheses: (-∞, ∞), because infinity is not attained.
  • Important caveat when using infinity in practice:
    • In MyLab Math/iCollege homework, you should not add a plus sign in front of ∞ (i.e., +∞); it may be treated as incorrect.
  • Domain determination via algebraic forms
    • For a rational function, exclude x values that make the denominator zero.
    • For a radical with even index (e.g., √(…) with index 2), exclude x values that make the radicand negative.
    • Real-world example restrictions: number of books bought cannot be negative; radius of a circle must be nonnegative; zero radius yields no circle; practical constraints apply in modeling.
  • Connect to the previous lecture: the domain of a function is the set of all x-values for which f(x) is defined; the range is the set of all possible output values y.

1.2: Graph of a Function

  • Graphing on the x-y plane

    • The graph of y = f(x) is the set of all points (x, y) where y = f(x).

  • Vertical Line Test (VLT)

    • A graph represents a function only if any vertical line x = c intersects the graph at most once.
    • If a vertical line crosses the graph in more than one point, the relation is not a function.

  • Intercepts (from the graph)

    • x-intercept(s): points where the graph crosses the x-axis (y = 0). These are solutions to f(x) = 0.
    • y-intercept: the point where the graph crosses the y-axis (x = 0). That value is f(0).
    • When a question mentions zeros or intercepts, they refer to x-intercepts where f(x) = 0 and y-intercept(s) corresponding to f(0).

  • Example interpretation from a sample graph

    • If a graph crosses the x-axis at x = 0 and x = 2π, then the domain on the graph is from x = 0 to x = 2π (if the graph only exists on that interval) and the corresponding y-values fall within the graph’s range on that interval.
    • Domain from 0 to 2π is written as

    [0, ext{ } 2 ext{ } ext{π}]

    • Range from -2 to 2 (inclusive) would be

    [−2,2].[-2, 2].

  • Reading intervals on a graph vs. an equation

    • If no graph is provided, determine domain by excluding invalid x-values (denominator zero, radicand negative for even roots) and including all other real x-values that yield a defined f(x).

  • Desmos as a visualization tool is recommended for exploring graphs interactively.

1.3: Properties of Functions

  • Even and odd functions

    • Even function: f(-x) = f(x) for all x in the domain. Graphically, symmetry about the y-axis.
    • Example: f(x) = x^2. Domain: (-∞, ∞). Indeed, f(-x) = (-x)^2 = x^2 = f(x).
    • Odd function: f(-x) = -f(x) for all x in the domain. Graphically, symmetry about the origin (rotate 180° about the origin).
    • Example: f(x) = x. Then f(-x) = -x = -f(x).

  • Symmetry references

    • Symmetry with respect to the x-axis would imply a non-function in general (horizontal symmetry can produce two y-values for a given x), so it is typically not a property of a function.
    • Symmetry about the origin corresponds to odd functions.

  • Monotonicity: increasing, decreasing, and constant on intervals

    • Increasing on an interval: as x increases, f(x) increases.
    • Decreasing on an interval: as x increases, f(x) decreases.
    • Constant on an interval: f(x) stays the same as x increases.
    • Example description from a sample graph:
    • Decreasing on
      (−6,−4)(-6, -4)
    • Increasing on
      (−4,0)(-4, 0)
    • Constant on
      [0,3][0, 3]
    • Decreasing on
      (3,6].(3, 6].

  • Local vs absolute extrema

    • Local maximum: a point where f(c) is greater than or equal to f(x) for x in some open interval around c.
    • Local minimum: a point where f(c) is less than or equal to f(x) for x in some open interval around c.
    • Absolute (global) maximum/minimum: the largest/smallest value of f on the entire domain.
    • Important distinction: a graph can have local extrema without having the same point as an absolute extremum.

  • Average rate of change and the secant line

    • Average rate of change of f from a to b:

    f(b)−f(a)b−a.\frac{f(b) - f(a)}{b - a}.

    • This value equals the slope of the secant line through points $(a,f(a))$ and $(b,f(b))$ on the graph.
    • Note on signs and ordering: if you write the numerator with f(a) first, you must maintain the corresponding denominator order (i.e., using a on the bottom if you started with a on top).

  • Practice example (recap from the session): reading a sample function's points and constructing a quick interpretation of its domain and range from a graph (e.g., domain from 0 to 2π; range from -2 to 2).

  • Desmos and digital tools

    • Desmos is recommended for interactive exploration of graphs and to visually confirm properties like symmetry, extrema, and intercepts.

Practical Question: Worked Example Set (from the end of the session)

  • Function 1: f(x) = (x - 3) / (x + 1)

    • Check if the point (5, 2) lies on the graph: compute f(5):

    f(5)=5−35+1=26=13.f(5) = \frac{5 - 3}{5 + 1} = \frac{2}{6} = \frac{1}{3}.

    • Since f(5) ≠ 2, the point (5, 2) is not on the graph.

  • Intercepts for f(x) = (x - 3) / (x + 1)

    • X-intercept: solve f(x) = 0. Set numerator to zero: x - 3 = 0 ⇒ x = 3. Check denominator nonzero: x + 1 ≠ 0 ⇒ x ≠ -1. So x-intercept: (3, 0).
    • Y-intercept: plug x = 0: f(0) = (0 - 3) / (0 + 1) = -3. So y-intercept: (0, -3).

  • Difference quotient for f(x) = (x - 3) / (x + 1)

    • Objective: compute

    f(x+h)−f(x)h.\frac{f(x+h) - f(x)}{h}.

    • Compute f(x+h) = rac{(x+h) - 3}{(x+h) + 1} = rac{x + h - 3}{x + h + 1}.
    • Then

    f(x+h)−f(x)h=x+h−3x+h+1−x−3x+1h.\frac{f(x+h) - f(x)}{h} = \frac{\frac{x + h - 3}{x + h + 1} - \frac{x - 3}{x + 1}}{h}.

    • Put over a common denominator: the common denominator is
      (x+h+1)(x+1).(x + h + 1)(x + 1).
    • After simplification (carrying through the algebra), one obtains:

    f(x+h)−f(x)h=4(x+1)(x+h+1).\frac{f(x+h) - f(x)}{h} = \frac{4}{(x + 1)(x + h + 1)}.

    • Quick check with a numerical example (e.g., x = 2, h = 0.1) yields approximately 0.430, consistent with the formula.

  • Notes on the teacher’s live session details (course logistics) from the transcript

    • Course format: online synchronous course (MyLab Math via iCollege) with weekly rhythm.
    • Schedule: two classes per week — Tuesday (lecture) and Thursday (breakout/lab session).
    • Attendance: attendance verification is crucial in the first two weeks; use your real student accounts; if you log in with a Gmail or non-GSU account, email the instructor with your name and the account used so attendance can be recorded.
    • Office hours: breakout session also serves as office hour; contact via GSU email or iCollege messaging.
    • MyLab Math access: three options exist to access Pearson MyLab Math; the recommended route is through iCollege to get the lowest price and integrated access; if using the bookstore, a course ID might be required (not always needed when linked via iCollege according to the lecturer).
    • Linking accounts: ensure your Pearson/MyLab Math account is linked to your iCollege account to access all quizzes, homework, and tests in one place.
    • Orientation quiz: an orientation quiz on MyLab Math must be completed with full credits to proceed to other quizzes and homework; the quiz details are located within the MyLab Math assignments on iCollege.
    • Recording and materials: the session is recorded by Webex and will appear on iCollege within about 24 hours; the PowerPoint might be uploaded there afterward.
    • Section notes: the instructor mentions two section identifiers (e.g., 11-11 and 0-99) and clarifies that this particular course uses the 11-11 section; attendance and course materials may differ for other sections.
    • Instructor and contact: Imo Su, PhD student in Georgia State University, Mathematics Department; contact via GSU email or iCollege messaging; the PowerPoint and materials will be posted on iCollege.

Summary and Connections

  • Core ideas connect through the concept of a function as a precise rule linking inputs to outputs, the graph as a visualization of that rule, and the analysis of how the function behaves (monotonicity, symmetry, extrema, and rate of change).
  • Practical tools (Desmos, Webex, iCollege, MyLab Math, Pearson) are introduced to support learning, attendance, and homework completion, with emphasis on linking accounts for lower costs and integrated access.
  • The session emphasizes careful notation (especially f(x) versus multiplication, interval notation, and the domain/range implications of algebraic expressions), and it provides concrete worked examples to reinforce these concepts.

End of notes