General Mathematics: Functions and Relations – Comprehensive Notes

Relation and Function: Core Concepts

• Relation: a set of ordered pairs; describes a connection between inputs (domain) and outputs (range).
• Function: a special type of relation where every input in the domain maps to exactly one output in the range.
• Domain: the set of all possible inputs (first elements) of the ordered pairs in a relation or function.
• Range: the set of all possible outputs (second elements) of the ordered pairs in a relation or function.
• Function as a machine: inputs → rule → outputs; emphasizes inputs, outputs, and the governing rule.
• Function machine components: inputs, outputs, and the rule that connects them.
• Well-defined rule: for every input there is a unique output.

Functions: Representations and Key Representations

• Function as a set of ordered pairs: f ⊆ (X × Y); a function requires that each x ∈ X appears with a unique y ∈ Y.
• Function as a table of values: table form f(x) = y lists input-output pairs; function if all x-values are distinct.
• Function as a mapping diagram: inputs on one side, outputs on the other, with arrows from x to its corresponding y.
• Function as a graph in the Cartesian plane: a graph representing the relation; visual check via vertical line test.
• Function defined by equation: a function where for each input x there is one output y determined by the equation.

Distinguishing a Function from a Non-Function (Examples and Practice)

• Example classification rule: a function requires exactly one output for every input.
• Function Machine A and B: outputs are single-valued for each input → function; Machine C has inputs with multiple outputs → not a function.

It's Your Turn: Practice on Function Machines

• A. Determine unknowns in the function machine setups.
• B. Identify which of the given relations are functions based on the mapping of inputs to outputs.
• C. Determine if given relations in a table are functions (ensure all x-values are unique in the table).
• D. Use a mapping diagram to decide if the relation is a function (check for unique y for each x).

Domain and Range: From a Function

• Domain: all real inputs allowed by the function (values of x for which the function is defined).
• Range: all possible outputs produced by the function (values of y).
• General rules (typical domains/ranges by function type):
  • Polynomial functions (linear, quadratic, cubic, etc.): domain is all real numbers
    $D = {x \mid x \in \mathbb{R}}$
  • Square root function: radicand must be ≥ 0 → domain constrained by the root; range typically ≥ 0 for the principal root
  • Rational function (fraction form): denominator ≠ 0 → x ≠ value that makes denominator 0; range may exclude certain outputs depending on the equation
  • Linear function: domain is all real numbers; range is all real numbers: $D = {x \mid x \in \mathbb{R}},\quad R = {y \mid y \in \mathbb{R}}$
  • Quadratic form $y = (x - h)^2 + k$ has the vertex at (h, k); range depends on the sign of the leading coefficient (a > 0 → y ≥ k; a < 0 → y ≤ k)
  • Square root specific: for $y = \sqrt{x} - c$, domain requires x ≥ 0; range starts from −c and upward
  • Rational function: ensure denominator ≠ 0; outputs depend on the specific function

Domain and Range: Worked Examples (Implicit from Transcript)

• Example 1: Find domain and range of $3x - y = 7$
  • Solve for y: $-y = -3x + 7 \Rightarrow y = 3x - 7$
  • Since this is linear in both x and y, the domain is all real numbers and the range is all real numbers.
  • Domain: $D = {x \mid x \in \mathbb{R}}$; Range: $R = {y \mid y \in \mathbb{R}}$
• Example 2: Find domain and range of $y = x^2 - 5$
  • Domain: all real numbers $D = {x \mid x \in \mathbb{R}}$
  • Range: since the parabola opens upward, $y \ge -5$ → $R = {y \mid y \ge -5}$
• Example 3: Find domain and range of $x^2 + y = 6$
  • Solve for y: $y = -x^2 + 6$
  • Domain: all real numbers $D = {x \mid x \in \mathbb{R}}$
  • Range: since the parabola opens downward, $y \le 6$ → $R = {y \mid y \le 6}$
• Example 4: Find domain and range of $y = \sqrt{x} - 3$
  • Domain (per transcript): $x \ge 3$; Range: $y \ge 0$
  • Note: Standard domain is $x \ge 0$ with range $y \ge -3$ (transcript appears to contain a misstatement here). The intended corrected form is: Domain $D = {x \mid x \ge 0}$; Range $R = {y \mid y \ge -3}$
• Example 5: Find domain and range of $4y + xy = 3x - 5$
  • Solve for y: $y(4 + x) = 3x - 5 \Rightarrow y = \dfrac{3x - 5}{4 + x}$
  • Domain: set of x for which the denominator is nonzero: $4 + x \neq 0 \Rightarrow x \neq -4$
  • Range: since this is a rational function with a variable in numerator and denominator, the transcript notes that the range excludes a particular value: $y \neq 3$ (i.e., 3 is not attained by the function). Thus, $D = {x \mid x \in \mathbb{R}, x \neq -4}$; $R = {y \mid y \in \mathbb{R}, y \neq 3}$
• Example 6: Find domain and range of $y = \dfrac{2x + 3}{x - 4}$
  • Domain: $x \neq 4$
  • Range: not all y values are attained; transcript states the range is all real numbers except $2$ → $R = {y \mid y \in \mathbb{R}, y \neq 2}$

Domain and Range from Graphs

• How to read a graph for domain and range:
  • Domain: all x-values shown on the graph (inputs)
  • Range: all y-values shown on the graph (outputs)
  • Notation conventions: bracket [ ] indicates included points; parentheses ( ) indicates excluded points and extends to infinity
• Examples (from transcript):
  • Example 1: Domain and Range are both all real numbers → $D = (-\infty, \infty),\quad R = (-\infty, \infty)$
  • Example 2: Domain all real; Range a single value 3 → $D = (-\infty, \infty),\quad R = [3]$
  • Example 3: Domain all real; Range a single value -1 → $D = (-\infty, \infty),\quad R = [-1]$
  • Example 4: Domain $D = [-1, \infty)$; Range $R = [1, \infty)$
  • Example 5: Domain $D = (-\infty, 3)$; Range $R = (0, \infty)$
  • Example 6: Domain $D = [-5, 2)$; Range $R = (-3, 3]$

Domain and Range from Piecewise Graphs

• For piecewise graphs, determine domain from the left-most x-value and range from the bottom y-value, then combine domains/ranges from all pieces.
• Examples (transcript):
  • Example 7: Domain $D = (-\infty, 4) \cup [5, \infty)$; Range $R = [-1, 3) \cup [4, \infty)$
  • Example 8: Domain $D = (-\infty, 1] \cup (2, \infty)$; Range $R = (-\infty, -1] \cup (-1, \infty)$
  • Example 9: Domain $D = (-\infty, -6] \cup (-5, 5)$; Range $R = (-\infty, 1] \cup (3, 5]$

Practice: Domain and Range from Function or Graph

• Function domain-range from a function:
  • 1) $y = -2x + 3$
  • 2) $y = (x - 2)^2 - 3$
  • 3) $y = 2x^2 - 1$
  • 4) $y = \dfrac{1}{x + 2}$
  • 5) $y = \sqrt{2x + 1}$
• Domain and range from graphs (including piecewise):
  • 6–10 (practice items corresponding to graphs not shown in transcript)

Functions as Real-Life Representations

• Functions model real situations; standard functional forms:
  • Linear: $f(x) = ax + b,\quad a \neq 0$
  • Quadratic: $f(x) = ax^2 + bx + c,\quad a \neq 0$
  • Polynomial of degree n: $f(x) = a<em>n x^n + a</em>{n-1} x^{n-1} + \cdots + a<em>1 x + a</em>0,\quad a_n \neq 0$
• Examples given in transcript:
  • Example 1: Cost of x meals at P50 each: $C(x) = 50x$
  • Example 2: Height as a function of age if height increases by 3 inches per year: $H(a) = 3 + a$ (note: transcript indicates 3 inches per year; typical interpretation would be $H(a) = 3a + H_0$, but transcript states the former)
  • Example 3: Battery charge B(h) with 11% loss per hour: $B(h) = 100 - 0.11h$
  • Example 4: Volume of a box created by cutting squares from a 12 in by 9 in rectangle: with L = 12 - 2x, W = 9 - 2x, h = x; $V(x) = (12 - 2x)(9 - 2x)x = 4x^3 - 42x^2 + 108x$

Piecewise Functions: Real-Life Examples

• Piecewise functions use more than one formula with different domains.
• Notation: $f(x) = { f<em>1(x) \text{ if } x \in \text{dom}</em>1,\; f<em>2(x) \text{ if } x \in \text{dom}</em>2, \; f<em>3(x) \text{ if } x \in \text{dom}</em>3, }$
• Examples given:
  • Example 5: Mobile plan t(n) with 0 < n ≤ 150 costing 350, and n > 150 costing 350 + (n - 150)
  • Example 6: Jeepney fare F(d) with 0 ≤ d ≤ 4 costing 13, and d > 4 costing 13 + 2.50(d - 4)
  • Example 7: Chocolate bars C(b) with 0 < b ≤ 5 priced at 65 each; b > 5 priced at 52 each

Constructing Functions: Practice Problems

• Tasks to construct functions from real-world descriptions:
  1) Bike rental: flat fare 568 plus 284 per hour for h hours → $C(h) = 568 + 284h$
  2) Jane’s earnings: 1022 per hour → $P(x) = 1022x$
  3) Rectangle with fixed perimeter 40 m; width x; express area A(x): L = (40/2) - x = 20 - x; hence $A(x) = x(20 - x) = 20x - x^2$
  4) Open-top box from 30 cm by 20 cm cardboard by cutting squares of side x: volume $V(x) = x(30 - 2x)(20 - 2x)$
  5) Ball thrown upward with initial velocity 24 m/s; height under gravity g = -4.9 m/s^2: $h(t) = h_0 + 24t - 4.9t^2$
  6) Taxi fare piecewise: first 2 km costs 227, plus 142 per km beyond 2 km: $F(d) = \begin{cases}227, &amp; 0 \le d \le 2 \ 227 + 142(d - 2), &amp; d &gt; 2\end{cases}$
  7) Parking lot: 170 for first 2 hours; 113 per hour after: $C(h) = \begin{cases}170, &amp; 0 \le h \le 2 \ 170 + 113(h - 2), &amp; h &gt; 2\end{cases}$
  8) Online store shipping: 284 per kg if w < 1; 454 per kg for 1 ≤ w ≤ 3; 681 per kg for w > 3: $S(w) = \begin{cases}284w, &amp; 0 < w < 1 \ 454w, & 1 \le w \le 3 \ 681w, & w > 3\end{cases}$

Summary: Key Takeaways

• A relation is any set of ordered pairs; a function is a relation with exactly one output for each input.
• Function representations include: set of ordered pairs, table of values, mapping diagram, graph in Cartesian plane, and equation form.
• The vertical line test: a graph represents a function if and only if any vertical line intersects the graph at most once.
• A function defined by an equation yields one output per input; otherwise, it is not a function.
• Domain and range capture the permissible inputs and outputs of a function:
  • Domain: inputs; Range: outputs.
  • Common domain/range patterns: polynomials and linear functions typically have domain all real numbers; square roots constrain the domain; rational functions exclude inputs that make the denominator zero; quadratic functions have ranges bounded by their vertex depending on orientation.
• For graphs, domain and range are read from the x-axis and y-axis respectively; interval notation uses brackets [ ] to indicate included endpoints and parentheses ( ) to indicate excluded endpoints; infinity is denoted with the corresponding bracket/parenthesis convention.
• Piecewise functions use different formulas over different input domains; domain of the whole function is the union of the subdomains.
• Real-life modeling uses standard function types (linear, quadratic, polynomial) to represent costs, heights, battery life, volumes, etc.

References (as listed in transcript)

• Queaño, M. (2021). General Mathematics Quarter 1 – Module 1: Functions, Department of Education Senior High School.
• OpenLearn resource on domain and range; Banigon et al. (2016); Crisologo & Ocampo (2016); Study.com resources on functions and domain/range.
• Cuemath, Kox, and other provided online references on domain and range and piecewise functions.