General Mathematics: Functions and Domain/Range — Comprehensive Notes

Relation and Function: Foundations

A relation is any set of ordered pairs. The domain of a relation is the first element of each ordered pair, and the range is the second element. A function is a special kind of relation or rule of correspondence between two elements (domain and range) such that every input (domain element) maps to exactly one output (range element). A function can be viewed as a machine with inputs, a rule that connects inputs to outputs, and outputs. This machine perspective helps flexibly describe how a function operates across various representations.

How a Function is Represented

There are several equivalent ways to express a function.

Function as a Machine

A function machine emphasizes three aspects: inputs, outputs, and the rule that links them. The idea is to take an input, apply a rule (e.g., double, square, add a constant), and obtain an output. A single input must produce exactly one output for the mapping to be a function.

Function as a Set of Ordered Pairs

A function can be represented by a set of ordered pairs (x, y) where x is the input (domain) and y is the output (range). To be a function, each input x must appear with exactly one corresponding y in the set. Example: if input 1 corresponds to output 5, the ordered pair is (1, 5).

Function as a Table of Values

We can denote the function with f(x) = y for each ordered pair (x, y). The same information can be organized into a table, where the x-values (domain) map to y-values (outputs). A table represents a function when all x-values are distinct in each row set.

Function as a Mapping Diagram

A mapping diagram shows inputs on the left, outputs on the right, and arrows from each input to its output. A function diagram requires that each input has a unique arrow to exactly one output. If any input maps to more than one output, the diagram does not define a function.

Function as a Graph in the Cartesian Plane

A graph in the Cartesian plane can represent a relation or a function. The vertical line test is a quick way to check if the graph represents a function: a graph represents a function if and only if any vertical line drawn intersects the graph at most once.

Function Defined by an Equation

A function defined by an equation will have at most one output for any input. If there is more than one output for a given input, the equation does not define a function. To test this, one can solve for y (or x) and examine how many outputs exist for each input.

Key Question: Differences Between Vertical and Horizontal Line Tests

Vertical line test: used with graphs. If any vertical line intersects the graph more than once, the graph is not a function.
Horizontal line test: used to check if x ↦ y is a function of y in parametric or inverse contexts; it’s less common in basic function definitions but helps to analyze whether a graph passes a one-to-one correspondence when solving for x as a function of y.

Domain and Range: Definitions and Rules

Domain: the set of all possible inputs x for which the function is defined (the values x can take without causing undefined expressions).
Range: the set of all possible outputs y produced by the function.

Rules for finding domain and range (typical cases):

Polynomial functions (linear, quadratic, cubic, etc.): Domain is all real numbers, D = {x | x ∈ ℝ}.
Square root functions: The radicand must be ≥ 0, so the domain is determined by x ≥ 0 or the equivalent condition that the inner expression is nonnegative.
Rational functions (fractions): Denominator ≠ 0, so first identify where the denominator is zero and exclude those x values from the domain.
Linear functions: Domain is all real numbers; Range is all real numbers.
Quadratic functions: Depending on the form, the range is constrained by the leading coefficient and the vertex; e.g., y = (x − h)^2 + k has y ≥ k if a > 0 and y ≤ k if a < 0.
For square roots, the range is typically y ≥ 0 (or y ≤ 0 depending on the expression) due to the square root output being nonnegative (or nonpositive if multiplied by −1, etc.).
When solving for domain and range by algebraic manipulation, you can rearrange to express y in terms of x or x in terms of y and apply the same domain/range constraints, including the allowance or restriction of square roots and the nonzero condition in denominators.

Example patterns:

If you have the equation 3x − y = 7, you can solve for y to analyze the range or for x to analyze the domain. Since the expression is linear in both variables, the domain and range are all real numbers.
If you have y = x^2 − 5, the domain is all real numbers, and the range is y ≥ −5 (the parabola opens upward with vertex at (0, −5)).
If you have x^2 + y = 6, rewrite as y = −x^2 + 6. The domain is all real numbers; the range is y ≤ 6 (opens downward with vertex at (0, 6)).
If you have y = √x − 3, the domain is x ≥ 3, and the range is y ≥ 0.
If you have a linear combination with y in the numerator and a linear denominator, such as 4y + xy = 3x − 5, after solving for y you may obtain a rational form in terms of x and y; identify restrictions from the denominator (e.g., 4 + x ≠ 0) and from any remaining expressions to determine the range of y. In this example, y ≠ 3 emerges as a restriction on the possible y-values (to avoid division by zero in the rearranged expression).
For equations like y = (2x + 3)/(x − 4), the domain is x ≠ 4, and the range excludes y = 2 (since x(y − 2) = 4y + 3, which fails when y = 2).

Domain and Range from Graphs

Domain from a graph: all x-values shown on the graph (the horizontal axis values that appear in the graph).
Range from a graph: all y-values shown on the graph (the vertical axis values that appear in the graph).
Brackets [ ] indicate that the endpoint is included; parentheses ( ) indicate the endpoint is excluded; infinity is shown as the limit behavior.

Examples of domain and range from graphs:

Example 1: Domain D = (−∞, +∞), Range R = (−∞, +∞).
Example 2: Domain D = (−∞, +∞), Range R = [3].
Example 3: Domain D = (−∞, +∞), Range R = [−1].
Example 4: Domain D = [−1, +∞), Range R = [1, +∞).
Example 5: Domain D = (−∞, 3), Range R = (0, +∞).
Example 6: Domain D = [−5, 2), Range R = (−3, 3].

Domain and Range for Piecewise Graphs

To determine domain and range for piecewise graphs, start from the left side of the graph for domain and move upward from the bottom for range. Each piece has its own domain, and the overall domain/range is the union of those individual domains/ranges.
Example 7: Domain D = (−∞, 4) ∪ [5, +∞), Range R = [−1, 3) ∪ [4, +∞).
Example 8: Domain D = (−∞, 1] ∪ (2, +∞), Range R = (−∞, −1] ∪ (−1, +∞).
Example 9: Domain D = (−∞, −6] ∪ (−5, 5), Range R = (−∞, 1] ∪ (3, 5].

Applications: Functions as Representations of Real-Life Situations

Functions model real phenomena. Common types include linear, quadratic, and polynomial functions, with definitions:

Linear function: $f(x) = a x + b, ext{ where } a eq 0.$ Client scenarios include cost, salary, or distance-time relationships.
Quadratic function: $f(x) = a x^2 + b x + c, ext{ where } a eq 0.$ Used for area, projectile motion, etc.
Polynomial function of degree n: $f(x) = an x^n + a{n-1} x^{n-1} + ext{…} + a1 x + a0, ext{ where } a_n eq 0.$

Examples illustrating functional models:

Example: Cost of x meals at P50 per meal: $C(x) = 50x.$
Example: Height H as a function of age a, with height increasing by 3 inches per year: $H(a) = 3 + a.$ (Note: the text sometimes mentions 2 inches per year elsewhere; the provided example uses +3 per year with an assumed starting height.)
Example: Battery charge B as a function of hours h with 11% loss per hour: $B(h) = 100 - 0.11 h.$
Example: Volume V of a box formed by cutting squares from a 12 in by 9 in rectangle: cutting out x-by-x squares, then L = 12 − 2x, W = 9 − 2x, H = x, so
$V(x) = (12 - 2x)(9 - 2x)x = 4x^3 - 42x^2 + 108x.$
Example: Piecewise function for a mobile plan with 150 free texts and P1 per extra text:
t(n) = egin{cases} 350, & 0 < n ext{ ≤ } 150,\ 350 + (n - 150), & n > 150. \ ext{(units in pesos)} \ ext{(where } n ext{ is the number of messages).} \ ext{Remark: this illustrates how a price function can be piecewise.} \
ext{(Other similar piecewise cost examples include jeepney fare, chocolate pricing, etc.)}
\

ight.

Example: Jeepney fare as a piecewise function: for distance d, the fare is $F(d)=\begin{cases}13, & 0 \le d \le 4, \ 13 + 2.50(d-4), & d > 4.\ \end{cases}$
Example: Chocolate bar price with bulk discount: $C(b)=\begin{cases}65 b, & 0 5.\ \end{cases}$

Practice Problems and Exercises (Illustrative Tasks)

Determine which of the following describes a Function.
- A set of ordered pairs can define a function if each input maps to a single output. Examples include simple linear pairs and single-valued mappings.
- Given a list of candidate relations, check if each input value x appears with a unique output y.
Practice conversions between representations:
- Convert a relation given as a set of ordered pairs to a table of values and determine whether it is a function.
- Identify which mapping diagrams represent functions by checking for multiple outputs per input.
Domain and Range from Graph problems:
- Identify the domain from the x-values shown in the graph and the range from the y-values shown.
- For piecewise graphs, determine the domain and range by considering each piece and taking the union of the corresponding domains/ranges.

Tasks: Construct Functions from Scenarios

Use the following prompts to practice constructing functions:
1) A bike shop charges a flat fare plus an hourly rate. Construct a function C(h) for the total cost for renting a bike for h hours.
2) Jane earns a rate of money per hour. Construct a function P(x) for total earnings after x hours.
3) A rectangle has a fixed perimeter; express area A(x) as a function of width x.
4) A square with side length x is cut from a 30 cm by 20 cm rectangle to form an open-top box; construct V(x).
5) A ball thrown upward with initial velocity 24 m/s, height h(t) at time t given gravity g = −4.9 m/s^2; construct h(t).
6) A taxi charges a base fare for the first 2 km and a per-kilometer rate thereafter; construct a piecewise function d ↦ cost.
7) A parking lot charges a base time with additional per-hour charges after 2 hours; construct C(h).
8) A store charges different per-kilogram rates depending on weight; construct a piecewise S(w).

Summary of Core Concepts

A relation is any set of ordered pairs; a function is a relation where every input has exactly one output.
Function representations include: machine, set of ordered pairs, table, mapping diagram, graph, and equation.
The vertical line test determines whether a graph represents a function: each vertical line intersects at most once.
A function defined by an equation yields one output per input; if more than one output exists for some input, the equation does not define a function.
Domain is the set of all valid inputs; range is the set of all possible outputs. Specific rules apply depending on function form (polynomials, square roots, rational expressions).
Domain and range can also be found by algebraic manipulation (solving for y or x) or by analyzing the graph.
Piecewise functions use different formulas for different input intervals; their domains are the unions of the subdomain intervals for each piece.
Functions model real-life phenomena (cost, height, battery, volume, etc.) and can be extended to more complex relationships with piecewise or multi-variable considerations.

References and Further Reading

Queaño, M. (2021). General Mathematics Quarter 1 – Module 1: Functions. Department of Education Senior High School.
OpenLearn and additional sources cited in the original materials for domain and range details, and piecewise function analysis.
Study resources: domain and range from graphs and how to determine the domain given a function or graph.