Section 2.1 Basics of Functions and Their Graphs

Objectives

• Find the domain and range of a relation.

• Determine whether a relation is a function.

• Determine whether an equation represents a function.

• Evaluate a function.

• Graph functions by plotting points.

• Use the vertical line test to identify functions.

• Obtain information about a function from its graph.

• Identify the domain and range of a function from its graph.

• Identify intercepts from a function’s graph.

Relations and Functions

Definition of a Relation

A relation is any set of ordered pairs. The set of all first components (usually x-values) is called the domain of the relation, and the set of all second components (usually y-values) is called the range of the relation.

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

Example: Given the relation { (1, 2), (3, 4), (5, 6) }, the domain is {1, 3, 5} and the range is {2, 4, 6}.

Definition of a Function

A function is a correspondence from a first set (the domain) to a second set (the range) such that each element in the domain corresponds to exactly one element in the range.

• No two ordered pairs have the same first component with different second components.

Example: The relation { (2, 3), (4, 5), (2, 6) } is not a function because the input 2 is paired with both 3 and 6.

Determining Whether a Relation is a Function

• Check if every element of the domain is paired with exactly one element in the range.

• If any input value is paired with more than one output value, the relation is not a function.

Example: The relation { (1, 2), (2, 3), (3, 4) } is a function because each input has only one output.

Functions as Equations

When an equation is solved for y in terms of x, it defines y as a function of x if and only if for each x in the domain, there is exactly one corresponding y-value.

• If for some x-values, more than one y-value is possible, the equation does not define y as a function of x.

Example: The equation does not define y as a function of x because for x > 0, there are two possible y-values for each x (one positive and one negative).

Function Notation and Evaluation

Function Notation

Function notation uses symbols such as , read as "f of x" or "f at x", to represent the value of the function at the number x.

• replaces y and emphasizes the dependence of the output on the input x.

Example: If , then .

Evaluating a Function

• To evaluate a function at a specific value, substitute the value for x in the function's formula.

Example: If , then .

Graphs of Functions

Graphing Functions by Plotting Points

The graph of a function consists of all ordered pairs (x, f(x)). To graph a function:

1. Select several values for x within the domain.

2. Calculate the corresponding y-values using the function rule.

3. Plot the points (x, y) on the coordinate plane.

4. Connect the points smoothly if the function is continuous.

Example: For , plot points for and connect them to form a straight line.

Comparing Graphs of Related Functions

• Shifting a graph up or down corresponds to adding or subtracting a constant from the function.

• Example: If , the graph of g is the graph of f shifted down by 3 units.

The Vertical Line Test for Functions

The vertical line test is a graphical method to determine if a graph represents a function:

• If any vertical line intersects the graph at more than one point, the graph does not represent a function.

• If every vertical line intersects the graph at most once, the graph does represent a function.

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

Analyzing Graphs of Functions

Obtaining Information from a Function's Graph

• To find , locate x = a on the x-axis and find the corresponding y-value on the graph.

• To find for which x-values , look for points where the graph has y-coordinate b.

Identifying Domain and Range from a Graph

• The domain is the set of all x-values for which the graph exists (i.e., for which there are points on the graph).

• The range is the set of all y-values that the graph attains.

Example: If a graph extends from to , the domain is . If the lowest point is and the highest is , the range is .

Identifying Intercepts from a Function’s Graph

• x-intercepts: Points where the graph crosses the x-axis (set and solve for x).

• y-intercept: Point where the graph crosses the y-axis (set and solve for y).

• A function can have multiple x-intercepts but at most one y-intercept.

Example: For , the x-intercepts are found by solving (so and ), and the y-intercept is .

Summary Table: Key Concepts

Additional info: Some examples and solutions were inferred or expanded for clarity and completeness, as the original material referenced but did not fully display them.