Precalculus Study Notes: Functions, Graphs, and Algebraic Foundations

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Chapter 1: Functions & Graphs

Graphs & Graphing Utilities

The study of functions and their graphs begins with understanding the Cartesian (rectangular) coordinate system, which is fundamental for visualizing mathematical relationships.

Cartesian Plane: A two-dimensional plane defined by a horizontal x-axis and a vertical y-axis, intersecting at the origin (0,0).
Quadrants: The axes divide the plane into four regions called quadrants, numbered I to IV in a counterclockwise direction.
Ordered Pair: A point on the plane is represented as (x, y), where x is the horizontal coordinate and y is the vertical coordinate.
Graph of an Equation: The set of all points (x, y) that satisfy a given equation in two variables.

Example: Does (0,0) satisfy the equation ? Substitute x = 0, y = 0: , which is false. Thus, (0,0) is not on the graph.

x-intercept: The point(s) where the graph crosses the x-axis (set y = 0 and solve for x).
y-intercept: The point(s) where the graph crosses the y-axis (set x = 0 and solve for y).

Example: Find the x- and y-intercepts of .

For x-intercept:
For y-intercept:

Example: Find the intercepts of .

x-intercept: Set y = 0:
y-intercept: Set x = 0:

Point Plotting

To graph an equation, plot several points that satisfy the equation and connect them smoothly.

Chapter P: Prerequisites – Fundamental Concepts of Algebra

Algebraic Manipulation and Review

Algebraic skills are essential for solving equations, simplifying expressions, and preparing for more advanced topics.

Solving Linear Equations: Isolate the variable using inverse operations.
Quadratic Equations: Equations of the form can be solved by factoring, completing the square, or using the quadratic formula:

Evaluating Expressions: Substitute given values for variables and simplify.
Order of Operations: Follow PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

Example: Evaluate at :

Section 1.2: Basics of Functions & Their Graphs

Relations and Functions

A relation is any set of ordered pairs. A function is a special type of relation where each input (domain) corresponds to exactly one output (range).

Domain: The set of all possible input values (x-values).
Range: The set of all possible output values (y-values).
Independent Variable: Usually x; the input value.
Dependent Variable: Usually y; the output value, which depends on x.

Example: is a function because each x-value has exactly one y-value.

Example: is not a function because some x-values correspond to two y-values (e.g., gives and ).

Function Notation and Evaluation

Function Notation: , , , etc., where f, g, h are function names.
To evaluate, substitute the input value for x.

Example: If , then (as above).

Evaluating at Expressions: means substitute for x in the function.

Example:

Determining Functions from Graphs

Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.
Closed Circle: Indicates the endpoint is included in the graph.
Open Circle: Indicates the endpoint is not included.

Domain and Range: Notation and Determination

Interval Notation: Used to describe sets of numbers.
Closed Interval [a, b]: Includes endpoints a and b.
Open Interval (a, b): Does not include endpoints.
Set-Builder Notation: Describes a set using a property, e.g., .

Examples:

Domain: means x-values from -2 (included) up to but not including 2.
Domain: means x-values between 1 and 4, not including endpoints.
Range: means y-values from 1 to 5, inclusive.

Finding Domain Algebraically

Square Roots: The expression inside a square root must be non-negative.
Fractions: The denominator cannot be zero.

Example: For , domain is .

Table: Interval and Set-Builder Notation Comparison

Interval Notation	Set-Builder Notation	Description
[a, b]	{ x \| a ≤ x ≤ b }	All x between a and b, inclusive
(a, b)	{ x \| a < x < b }	All x between a and b, not including endpoints
[a, b)	{ x \| a ≤ x < b }	All x with a included, b not included
(-∞, b]	{ x \| x ≤ b }	All x less than or equal to b
(a, ∞)	{ x \| x > a }	All x greater than a

Additional info:

Students are encouraged to read the textbook before class and study at least two hours after class for mastery.
Daily assessments and quizzes are frequent; lowest two quiz grades are dropped, and the final exam can replace a missing exam grade.
Homework is assigned from Chapter P (Prerequisites) and Chapter 1, focusing on algebraic manipulation and graphing skills.