Section 1: Relations and Functions

Part 1: Relations

A relation is a correspondence between two variables: the domain (set X) and the range (set Y). In a relation, each element from the domain corresponds to at least one element from the range. The domain is often considered the set of inputs, while the range is the set of outputs. Relations can be described verbally, numerically, graphically, or algebraically.

• Verbal Description: Describes the relationship in words.

• Numerical Description: Uses tables or lists of ordered pairs to show the relation.

• Graphical Description: Plots ordered pairs on a coordinate plane or uses mappings.

• Algebraic Description: Uses equations to represent the relation.

Example: An astronaut throws a rock on the Moon and measures its height at various times. The domain is the set of times, and the range is the set of heights.

The same relation can be represented graphically or as a mapping diagram:

Example: Consider the relation {(0, 3), (-1, 4), (0, 5), (3, 0)}. The domain is {0, -1, 3}, and the range is {3, 4, 5, 0}.

Part 2: Functions

A function is a special type of relation in which each element of the domain corresponds to only one element of the range. All functions are relations, but not all relations are functions. Functions can be expressed verbally, numerically, graphically, or algebraically.

• To determine if a relation is a function, check that each input (domain value) has only one output (range value).

• Vertical Line Test: A graph represents a function if every vertical line intersects the graph at most once.

Example: The equation is a function, but is not, since some x-values correspond to two y-values.

Part 3: Function Notation

Functions are typically labeled with letters such as f and g. If is a function, then denotes the output for input . The notation means that is a function of $x$. To evaluate a function, substitute a specific value for and compute .

• Example: If , then .

Part 4: Domain of Functions

The domain of a function is the set of all possible input values (x-values) for which the function is defined. To find the domain, determine which x-values do not cause mathematical issues (such as division by zero or taking the square root of a negative number).

• Polynomial Functions: Domain is all real numbers, .

• Rational Functions: Domain excludes values that make the denominator zero.

• Radical Functions: For even roots, the radicand must be non-negative.

Example: For , the domain is .

Part 5: Function Equations

From the equation of a function, you can find:

• Intercepts: Points where the graph crosses the axes.

• x-intercept (zero): Set and solve for .

• y-intercept: Set and solve for .

Section 2: The Graph of a Function

Analyzing Graphs of Functions

The graph of a function consists of all ordered pairs plotted on a coordinate plane. From the graph, you can determine:

• Domain and Range: The set of all x-values and y-values represented on the graph.

• Intercepts: Where the graph crosses the axes.

• Intervals of Positivity/Negativity: Where (above x-axis) or (below x-axis).

• Increasing/Decreasing/Constant Intervals: Where the function rises, falls, or remains flat as x increases.

• Local and Absolute Extrema: Local maxima/minima are the highest/lowest points in a neighborhood; absolute extrema are the highest/lowest points on the entire domain.

Section 3: Graphing Functions

Part 1: Library of Functions

A family of functions is a group of functions with similar shapes and properties. The parent function is the simplest form in the family. Common parent functions include:

Part 2: Graphing Functions Using Transformations

Transformations allow you to graph new functions based on parent functions by shifting, stretching, compressing, or reflecting their graphs.

• Vertical Shifts: shifts up by units; shifts down by $k$ units.

• Horizontal Shifts: shifts right by units; shifts left by $h$ units.

• Vertical Stretch/Compression: stretches if , compresses if .

• Horizontal Stretch/Compression: compresses if , stretches if .

• Reflections: reflects over the x-axis; reflects over the y-axis.

Summary Table of Transformations:

Example: To graph , start with the parent function , shift the graph up by 3 units.

Additional info: These foundational concepts are essential for understanding more advanced topics in College Algebra, such as polynomial, rational, exponential, and logarithmic functions.