College Algebra: Functions, Graphs, and Set Notation Study Guide

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Introduction to Graphs and Functions

Understanding Graphs of Equations

The graph of an equation is the set of all ordered pairs (x, y) that satisfy the equation. Graphs help visualize changes and important features of equations. By plotting enough points, we can determine the overall shape of the graph.

Ordered Pair: A pair of numbers (x, y) representing a point on the coordinate plane.
Example: The point (0, 0) is at the origin; (5, 7) is 5 units right and 7 units up from the origin.

Graphing Equations by Plotting Points

Plotting Points

To graph an equation, select values for x, compute the corresponding y values, and plot the resulting points.

Example: For , choose x-values, calculate y, and plot the points to reveal a downward-opening parabola.
Example: For , plot points to reveal a V-shaped graph.

Intercepts

Intercepts are points where the graph crosses the axes.

x-intercept: Where the graph crosses the x-axis (y = 0).
y-intercept: Where the graph crosses the y-axis (x = 0).
Example: For , set y = 0 to find the x-intercept, and x = 0 to find the y-intercept.

Interpreting Information from Graphs

Reading and Using Graphs

Graphs can be used to interpret real-world data and answer questions about trends and relationships.

Example: A graph showing the probability of divorce by age at marriage can be used to estimate percentages for different years after marriage.
Application: Use the graph to estimate values not explicitly given by interpolating between points.

Set Notation and Interval Notation

Set Notation

Set notation is used to describe collections of numbers.

Union (): All elements in either set A or set B.
Intersection (): All elements in both set A and set B.
Example: , , , .

Interval Notation

Interval notation is a shorthand way to describe subsets of the real number line.

Graphical Representation	Set Builder Notation	Interval Notation	In Words
Arrow to the right from a (open circle)	{x \| x > a}	(a, ∞)	All real numbers greater than a
Arrow to the right from a (closed circle)	{x \| x ≥ a}	[a, ∞)	All real numbers greater than or equal to a
Arrow to the left from b (open circle)	{x \| x < b}	(-∞, b)	All real numbers less than b
Arrow to the left from b (closed circle)	{x \| x ≤ b}	(-∞, b]	All real numbers less than or equal to b
Line segment between a and b (open circles)	{x \| a < x < b}	(a, b)	All real numbers between a and b, not including a or b
Line segment between a and b (closed circles)	{x \| a ≤ x ≤ b}	[a, b]	All real numbers between a and b, including a and b

Functions and Their Properties

Definition of a Function

A function is a relation in which each input (x-value) corresponds to exactly one output (y-value).

Domain: The set of all possible input values (x-values).
Range: The set of all possible output values (y-values).
Example: The relation {(6,1), (6,2), (8,3), (9,4)} is not a function because the input 6 corresponds to two different outputs (1 and 2).

Equations Representing Functions

Functions can be represented by equations, tables, or graphs.

Example: models the percentage of first-year college women claiming no religious affiliation as a function of years after 1970.
Application: Substitute values for x to find the corresponding y-values for specific years.

Vertical Line Test

The vertical line test is used to determine if a graph represents a function. If any vertical line crosses the graph more than once, the graph does not represent a function.

Example: The graph of a circle fails the vertical line test and is not a function of x.

Evaluating Functions

Substituting Values

To evaluate a function, substitute the given value for the variable.

Example: If , then .
Example: If , then .

Polynomials: Adding, Subtracting, and Multiplying

Adding and Subtracting Polynomials

Combine like terms to add or subtract polynomials.

Example:

Multiplying Polynomials

Use the distributive property or FOIL method to multiply polynomials.

Example:
Example:
Example:

Summary Table: Set Notation and Interval Notation

Set Builder Notation	Interval Notation	Graphical Representation
{x \| x > a}	(a, ∞)	Arrow to the right from a (open circle)
{x \| x ≥ a}	[a, ∞)	Arrow to the right from a (closed circle)
{x \| x < b}	(-∞, b)	Arrow to the left from b (open circle)
{x \| x ≤ b}	(-∞, b]	Arrow to the left from b (closed circle)
{x \| a < x < b}	(a, b)	Line segment between a and b (open circles)
{x \| a ≤ x ≤ b}	[a, b]	Line segment between a and b (closed circles)

Key Terms and Concepts

Function: A relation where each input has exactly one output.
Domain: Set of all possible input values.
Range: Set of all possible output values.
Intercepts: Points where the graph crosses the axes.
Vertical Line Test: A test to determine if a graph is a function.
Set Notation: Describes sets using braces and conditions.
Interval Notation: Describes intervals on the real number line using parentheses and brackets.