Precalculus Notes: Functions, Domain and Range (Comprehensive)

Course structure and approach

  • This is a brand-new precalculus course; live lessons follow a consistent sequence.

  • The instructor records the session and outlines a four-part format: note-taking, a Q&A break, practice problems, and optional extra practice/question time after the main lesson.

  • A student (Cole) has prepared notes for the class; encouragement is given to utilize peer-made resources.

  • Tools: Desmos will be used frequently for graphing; the session emphasizes graphing as a key way to learn concepts.

  • Typical lesson timing mentioned: start around 11:00, with a 15-minute conclude window for main content, followed by optional extended practice.

  • The lesson emphasizes familiarity with three notations for domain and range, practicing both graphing and algebraic reasoning.

Key concepts: function, domain, and range

  • A function is a mathematical idea defined by a set of inputs (domain) and outputs (range).

  • In graphs, inputs are represented by X and outputs by Y.

  • Domain: the set of all possible X-values (inputs) that the function can take.

  • Range: the set of all possible Y-values (outputs) that the function can produce.

  • When graphing, domain corresponds to the horizontal axis (X-values) and range to the vertical axis (Y-values).

  • Visual intuition: the domain are all inputs that can be plugged into the function; range is all possible outputs you can get from those inputs.

Notation types for domain and range

  • Three types of notation, all expressing the same idea: 1) Inequality (algebraic) notation

    • Example: domain expressed with inequalities like -X

      • Example: the domain could be
        -\infty < x < \infty, which means all real numbers (the phrase "domain is all reals" is equivalent).

    • Range example: y≥3y \ge 3 (or equivalently y∈[3,∞)y \in [3, \infty) in interval notation).

    • Important concept: infinities are always written with open/rounded brackets: (−∞,∞)(-\infty, \infty).
      2) Interval notation

    • Domain example: (−∞,∞)(-\infty, \infty) (all real numbers).

    • Range example: [3,∞)[3, \infty) if 3 is included; use parentheses (⋅)( \cdot ) if not included.

    • Brackets denote inclusion; parentheses denote exclusion.
      3) Set-builder notation

    • Domain example: D=x∈RD = { x \in \mathbb{R} } (x is a real number, i.e., all real numbers).

    • Range example: R=y∈R∣y≥3R = { y \in \mathbb{R} \mid y \ge 3 } (the vertical bar means "such that").

    • Set-builder can express exceptions explicitly, e.g., "y is a real number but y ≥ 3".

  • Why learn all three forms? Quiz questions may present any of the three; being fluent in all helps recognition and conversion between representations.

Graph features and interpretation

  • Intercepts

    • X-intercept: where the graph crosses the X-axis (y = 0).

    • Y-intercept: where the graph crosses the Y-axis (x = 0).

  • Increasing/Decreasing/Constant

    • Read the graph from left to right (like a book) to determine where the function increases, decreases, or stays constant.

    • Increasing: y-values rise as x-values increase over an interval.

    • Decreasing: y-values fall as x-values increase over an interval.

    • Constant: y-values do not change over an interval (horizontal segment).

  • Sign of the function (positive/negative)

    • Positive region: above the X-axis (Y > 0).

    • Negative region: below the X-axis (Y < 0).

    • Quadrants where the graph lies indicate where the function is positive/negative:

    • Quadrants I and II: y > 0 (above X-axis).

    • Quadrants III and IV: y < 0 (below X-axis).

  • Asymptotes

    • Vertical asymptotes: lines x = c where the graph approaches ±∞ as x → c. They create domain gaps at those x-values.

    • Horizontal asymptotes: lines y = L where the graph approaches L as x → ±∞.

    • Remember: vertical asymptotes correspond to restrictions on the domain; horizontal asymptotes correspond to restrictions on the range as x grows without bound.

  • Maximums and minimums

    • Local maximums: high points where the graph rises to a peak and then falls.

    • Local minimums: low points where the graph falls to a valley and then rises.

    • A legitimate maximum/minimum should have the graph rise/fall around it (i.e., the surrounding values must be on both sides).

  • End behavior

    • As x → -∞ or x → ∞, examine what y (the function value) does: tends to ±∞, or approaches a finite value, or oscillates.

    • Examples discussed: depending on the graph, end behavior could be going to +∞, -∞, or approaching a finite horizontal asymptote.

  • Symmetry (even/odd/origin)

    • Even functions: symmetric about the y-axis; f(-x) = f(x).

    • Odd functions: symmetric about the origin; f(-x) = -f(x).

    • Origin symmetry (central symmetry): the graph looks the same after a 180° rotation about the origin.

    • Intuition: even functions have the same end behavior on both sides of the y-axis; odd functions have opposite end behavior across the origin.

Practice concepts and example scenarios (guidance from the lesson)

  • Example: Interpreting a graph to find intervals of increase/decrease

    • A sample graph may show:

    • Increasing on the interval from x = -3 to x = 0.

    • Decreasing on two intervals: from -∞ up to -3 and from x = 0 to ∞.

    • Domain notation for this example (in interval form):

    • Increasing: (−3,0)(-3, 0)

    • Decreasing: (−∞,−3)∪(0,∞)(-\infty, -3) \cup (0, \infty)

    • Important notes: endpoints like -3 and 0 are not included if the graph only shows the increase/decrease on open intervals; if those points are included by the graph, brackets would be used.

  • Example: Identifying asymptotes from graphs

    • Graph 1 (left) features multiple vertical asymptotes (example values given): at x = -2, x = 0, and x = 1; a horizontal asymptote at y = 1.

    • Graph 2 (right) features a horizontal asymptote at y = 0 and vertical asymptotes at certain x-values (examples discussed).

    • How to read them: vertical asymptotes are lines that the graph approaches but never reaches; horizontal asymptotes are constant y-values approached as x grows large in magnitude.

  • Example: Domain and range from a piecewise or multi-segment graph

    • For a left graph with a vertical asymptote at x = -2, the domain would exclude x = -2 and be split around it; for a graph with a second vertical asymptote at x = 0, there would be an additional domain split.

    • If a horizontal asymptote is at y = 3 in a certain scenario, then the range would be restricted accordingly (y cannot cross that asymptote as x → ±∞).

    • When a graph has two disjoint pieces, the domain is the union of the x-intervals covered by each piece: e.g., (−∞,−2)∪(−2,0)∪(0,∞)(-\infty, -2) \cup (-2, 0) \cup (0, \infty), etc., depending on the exact graph.

  • Example: End behavior notation practice

    • For a graph where as x → -∞, y → +∞ or −∞, record the appropriate end behavior (
      e.g., as x → -∞, f(x) → +∞; as x → ∞, f(x) → −∞, etc.).

  • Example: A path graph (hiker) domain interpretation

    • A path modeled from x = -10 to x = 10, with the path covering all those x-values, gives a domain of
      [−10,∞)[-10, \infty)? Actually, the example concluded the domain is [−10,10][-10, 10] with both endpoints included, since every x-value between -10 and 10 appears somewhere on the path.

    • Key takeaway: the domain is the set of x-values that actually occur on the graph; if every x between two endpoints occurs somewhere on the graph, the domain is the closed interval including both endpoints.

Formulas and symbols to remember (LaTeX format)

  • Endpoints and notations

    • Interval notation for all real numbers: (−∞,∞)(-\infty, \infty)

    • Range with inclusive endpoint: [a,∞)[a, \infty); with exclusive endpoint: (a,∞)(a, \infty)

  • Set-builder conventions

    • Domain: D=x∈RD = { x \in \mathbb{R} }

    • Range: R=y∈R∣y≥3R = { y \in \mathbb{R} \mid y \ge 3 }

  • Function relationships

    • Intercepts: x-intercept where f(x)=0f(x) = 0; y-intercept at x=0x = 0, value is f(0)f(0)

    • Even function: f(−x)=f(x)f(-x) = f(x)

    • Odd function: f(−x)=−f(x)f(-x) = -f(x)

  • End behavior and asymptotes

    • Vertical asymptote: line x=cx = c where the graph tends to ±∞ as x → c

    • Horizontal asymptote: line y=Ly = L where f(x)→Lf(x) \to L as x→±∞x \to \pm\infty

  • Slope and rate of change

    • Slope (rate of change) of a line: m=ΔyΔxm = \frac{\Delta y}{\Delta x}, often written as m=riserunm = \frac{\text{rise}}{\text{run}}; the symbol Δ\Delta represents a change in a quantity.

Practical tips for studying and exam prep

  • Always read graphs left to right to identify increasing/decreasing behavior and to locate intercepts and asymptotes.

  • Be comfortable converting between notation forms: inequality form, interval form, and set-builder form.

  • Learn to recognize when endpoints are included (brackets) vs excluded (parentheses).

  • Practice identifying domain and range on graphs with multiple pieces and with asymptotes; remember that vertical asymptotes create domain gaps, horizontal asymptotes constrain the range at infinity.

  • Remember the definitions and examples for even/odd/origin symmetry; practice by testing f(-x) relative to f(x).

  • Use Desmos as a visualization aid to confirm your interval conclusions and end behavior reasoning.

Quick reference: summary of key ideas

  • Domain = set of x-values the function takes; range = set of y-values produced.

  • Notation recap:

    • Inequality: e.g., x∈(−∞,∞)x \in (-\infty, \infty); y≥3y \ge 3

    • Interval: e.g., Domain (−∞,∞)(-\infty, \infty); Range [3,∞)[3, \infty)

    • Set-builder: e.g., D=x∈RD = { x \in \mathbb{R} }; R=y∈R∣y≥3R = { y \in \mathbb{R} \mid y \ge 3 }

  • Graph features to locate quickly:

    • X- and Y-intercepts

    • Intervals of increase/decrease, and constant segments

    • Regions where the graph is positive/negative

    • Vertical/horizontal asymptotes and their implications for domain/range

    • Local maximums/minimums

    • End behavior as x→±∞x \to \pm\infty

    • Symmetry: even, odd, origin (central) symmetry

  • Example-driven strategies: identify precise x-values where behavior changes; note inclusivity using brackets vs parentheses; translate between representations as needed.

Final study tips from the session

  • Expect to encounter all three types of notation in quizzes; practice converting between them.

  • When given a graph, extract domain and range by inspecting the visible x/y-values and the presence of asymptotes.

  • Practice with lines of best-fit slope intuition to understand rate of change (rise over run).

  • Use extra practice problems if you need more drill on a particular topic before an assessment.

  • If something feels like a new language at first (e.g., set-builder notation), keep tracing the meaning with simple examples and compare to interval notation for clarity.