Chapter 2: Functions and Graphs

Section 2.1: Basics of Functions and Their Graphs

This section introduces the foundational concepts of functions, their graphical representations, and essential techniques for analyzing and interpreting functions in algebra and precalculus.

Definition of a Relation

• Relation: Any set of ordered pairs. The domain is the set of all first components (inputs), and the range is the set of all second components (outputs).

• Example: For 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

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

• Key Point: No two ordered pairs in a function have the same first component with different second components.

• Example: The relation {(2, 3), (4, 5), (2, 6)} is not a function because 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 only one element in the range.

• Example: {(1, 2), (2, 3), (3, 4)} is a function; {(1, 2), (1, 3)} is not.

Functions as Equations

• If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not define y as a function of x.

• Example: does not define y as a function of x because for some values of x (e.g., x = 4), there are two possible values for y (2 and -2).

Function Notation

• Notation: is read as "f of x" and represents the value of the function at the number x.

• Example: If , then .

Evaluating a Function

• To evaluate a function, substitute the given value into the function's formula.

• Example: If , then .

Graphs of Functions

• The graph of a function consists of all ordered pairs plotted in the coordinate plane.

• Example: For , plot points for several values of and connect them to form the graph.

Graphing Functions by Plotting Points

• Select integer values for , compute corresponding values, plot the points, and connect them smoothly.

• Example: For , plot , , , , .

Transformations of Graphs

• Shifting a graph up, down, left, or right changes the function's output or input accordingly.

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

The Vertical Line Test for Functions

• If any vertical line intersects a graph in more than one point, the graph does not define as a function of .

• Example: The graph of a circle fails the vertical line test, so it is not a function.

Obtaining Information from a Function's Graph

• Use the graph to find function values, intercepts, domain, and range.

• Example: To find , locate on the x-axis and read the corresponding value on the graph.

Identifying Domain and Range from a Function’s Graph

• Domain: All -values for which the graph exists (project the graph onto the x-axis).

• Range: All -values that the graph attains (project the graph onto the y-axis).

• Example: If a graph extends from to , the domain is .

Identifying Intercepts from a Function’s Graph

• x-intercepts: Points where the graph crosses the x-axis ().

• y-intercept: Point where the graph crosses the y-axis (). A function can have more than one x-intercept but at most one y-intercept.

• Example: For , the x-intercepts are at and ; the y-intercept is at .