Notes: Order of Operations, Domain & Range, Increasing/Decreasing, and Average Rate of Change

Order of Operations (PEMDAS) and Function Basics

• PEMDAS acronym stands for the order of operations:
  • P = Parentheses
  • E = Exponents
  • MD = Multiplication and Division (from left to right)
  • AS = Addition and Subtraction (from left to right)
• Example from slides to illustrate PEMDAS:
  • Expression: $-10 + \left[(11 - 5) + 3(-4)^2\right]$
  • Step 1 (Parentheses): $(11 - 5) = 6$
  • Step 2 (Exponents): $(-4)^2 = 16$; then $3\cdot 16 = 48$
  • Step 3 (Inside the bracket): $6 + 48 = 54$
  • Step 4 (Addition/Subtraction left to right): $-10 + 54 = 44$
• Emphasized rule: Follow PEMDAS to avoid order mistakes; the final result of the example is $44$.
• Practical takeaway: Always show the full step-by-step work when solving multi-step expressions.

Domain and Range Overview

• Domain: The set of all x-values for which a function is defined.
• Range: The set of all y-values produced by the function.
• X-axis and Y-axis terminology:
  • X-axis refers to the horizontal axis (input values).
  • Y-axis refers to the vertical axis (output values).
• Restricted domain / restrictions: Situations where the domain must be limited (e.g., square roots of negative numbers, denominators equal to zero).
• Domain and Range as concepts:
  • Domain is about where the function can take input values.
  • Range is about what outputs are produced.
• How to represent domain and range:
  • Interval notation, set-builder notation, and inequality notation.
  • For interval notation, use [ ] when an endpoint is included and ( ) when it is not.
  • For set-builder notation, use { x | condition }.
  • For inequality notation, express the bounds with relational symbols directly.
• Practical note: Arrows on a graph indicate the function continues indefinitely in that direction (unbounded domain or range).

Notation for Domain and Range (Different Representations)

• Finite interval notation:
  • Example: domain:
    -3 ≤ x ≤ 2
  • Interval notation: [−3, 2]
• Inequality notation:
  • Example: −3 ≤ x ≤ 2 (read as “x is between −3 and 2, inclusive”).
• Set notation:
  • Example: { x | −3 ≤ x ≤ 2 }
  • Example (open/closed variants): { x | −3 < x ≤ 2 }
• Interval notation examples:
  • Domain: D = [−3, 2]
  • Domain: D = (−3, 2]
• Range notation follows the same rules as domain (interval, set, or inequality form).

Domain and Range in Practice (Examples from Slides)

• Domain and Range forms (from slides):
  • Domain (interval notation):
  • (-5, 5]
  • Domain (set notation):
  • { x | −5 < x ≤ 5 }
  • Range (interval notation):
  • (-2, 2]
  • Range (set notation):
  • { y | −2 < y ≤ 2 }
• Unbounded domains (arrows on graphs):
  • Domain (interval notation): (−∞, ∞)
  • Domain (inequality notation): −∞ < x < ∞
  • Range continues similarly to infinity (e.g., y ∈ [a, ∞) or (−∞, b]).

Different Notations in Detail

• Set-builder notation: { x | condition about x }
  • Example: { x | −3 ≤ x ≤ 2 }
• Interval notation: D = [−3, 2] or D = (−3, 2], etc.
• Inequality notation: −3 ≤ x ≤ 2 or −3 < x ≤ 2
• Graphical note: The choice of open vs closed endpoints corresponds to whether the endpoint is included in the domain or range.

How to Read Endpoints on Graphs (Open vs Closed)

• Basic idea:
  • Open circle or parentheses ( ) indicate endpoints not included (open).
  • Closed circle or brackets [ ] indicate endpoints included (closed).
  • This translates to inequalities with strict (
• Examples:
  • -3 ≤ x ≤ 2 corresponds to interval [−3, 2].
  • -3 < x ≤ 2 corresponds to interval (−3, 2].
• Relationship between graph and notation:
  • Closed endpoint on a graph implies the corresponding bracket in interval notation is closed.
  • Open endpoint on a graph implies the corresponding bracket is open.

Domain/Range with Infinity and Inequality Form (Advanced)

• Examples:
  • Domain: (−∞, ∞) corresponds to all real numbers.
  • Range with a lower bound: { y | y ≥ −4 } corresponds to [−4, ∞).
• Notation of infinity:
  • Used with parentheses for open bounds: (−∞, a) or (a, ∞) depending on the endpoint inclusivity.
• Arrow notation on graphs communicates unboundedness in the indicated direction.

LCD and Adding Fractions (Concepts from Slides)

• LCD = Least Common Denominator (also called least common multiple for denominators when adding fractions).
• DESMOS note: You can use DESMOS as an alternative to graph or compute domain/range.
• Example (LCD): If denominators are 3 and 6, LCD = 6.
  • Convert 1/3 to 2/6, and express the other fraction with denominator 6 as needed: sum becomes (2 + 3x^2)/6 (illustrative for the slide example).

Solving for a Variable and Domain Restrictions (Algebraic Practice)

• Example: F = \frac{1}{2} s h
  • Solve for h: Multiply both sides by 2: 2F = s h
  • Isolate h: $h = \frac{2F}{s}$
  • Domain restriction: Denominator cannot be 0, so $s \neq 0$. Also, depending on context, F and h can be any real numbers as long as s ≠ 0.

Increasing and Decreasing Functions (Conceptual)

• Definitions (without derivatives, as introduced):
  • A function is increasing on an interval if, for any x1 < x2 in the interval, f(x1) < f(x2).
  • A function is decreasing on an interval if, for any x1 < x2 in the interval, f(x1) > f(x2).
• How to determine intervals from a graph or a derivative:
  • Look for where the function rises (increasing) or falls (decreasing) as x increases.
• Examples from slides (intervals given):
  • Example 1: Increasing on (−∞, 1], Decreasing on [1, ∞).
  • Example 2: Increasing on (−∞, −0.7] ∪ [0, 0.7], Decreasing on [−0.7, 0] ∪ [0.7, ∞).
  • Note: The first example may reflect a typical piecewise function with a turning point at x = 1; the second example shows multiple monotonic pieces with turning points at −0.7, 0, and 0.7.

Average Rate of Change (ARC/ROC)

• Definition:
  • The average rate of change of a function f on the interval [a, b] is
    $\text{ROC} = \frac{f(b) - f(a)}{b - a}.$
• Worked examples from slides:
  • Example 1: f(x) = x^2 - 7 on [2, 5]
  • Compute: f(2) = 4 - 7 = −3, f(5) = 25 - 7 = 18
  • ROC = \frac{18 - (−3)}{5 - 2} = \frac{21}{3} = 7
  • Example 2: f(x) = x^3 - 4x on [1, 3]
  • Compute: f(1) = 1 - 4 = −3, f(3) = 27 - 12 = 15
  • ROC = \frac{15 - (−3)}{3 - 1} = \frac{18}{2} = 9
• Step-by-step reminders:
  • Substitute a and b into f to find f(a) and f(b).
  • Then apply the ROC formula.

Quick Reference: Reminders for the Quiz

• Topics covered on the quiz (tomorrow):
  • Solving Multi-Step Equations (with PEMDAS)
  • Domain and Range concepts and notation
  • Increasing/Decreasing sections
  • Average Rate of Change calculations
• Tools mentioned:
  • DESMOS for graphing and quick checks
  • LCD (Least Common Denominator) for adding fractions

Practical Tips (From Slides)

• Always show work to earn credit on quizzes and assignments.
• Use DESMOS to visualize domain, range, and function behavior when possible.
• Remember the meanings of open/closed endpoints and how they map to interval notation.
• For domain restrictions, identify where the expression is undefined (division by zero, even roots of negative numbers, etc.).
• For rate of change problems, substitute the interval endpoints into the function first, then apply the ROC formula.

Quick glossary of terms introduced

• Domain: set of all possible x-values for which the function is defined.
• Range: set of all possible y-values produced by the function.
• X-axis: horizontal axis (input domain values).
• Y-axis: vertical axis (output range values).
• Restricted domain/restrictions: limitations on the input values due to the function’s definition.
• PEMDAS: order of operations (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).
• Average rate of change (ARC/ROC): the slope of the secant line over an interval, (\dfrac{f(b)-f(a)}{b-a}).
• LCD: least common denominator used to add fractions with different denominators.
• DESMOS: an online graphing calculator useful for visualizing functions and domains/ranges.

End of Notes