Section 1.2 Basics of Functions and Their Graphs

Introduction

This section introduces the foundational concepts of functions and their graphical representations, which are essential for understanding higher-level mathematics in Precalculus. Students will learn to identify, evaluate, and graph functions, as well as analyze their properties using various methods.

Definition of a Relation

• Relation: Any set of ordered pairs. Each ordered pair consists of a first component (input) and a second component (output).

• Domain: The set of all first components (x-values) of the ordered pairs.

• Range: The set of all second components (y-values) of the ordered pairs.

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

• 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.

• 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 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 only one element in the range.

• If any input corresponds to more than one output, the relation is not a function.

Example: The relation { (1, 2), (2, 3), (1, 4) } is not a function because 1 is paired with both 2 and 4.

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

Function Notation

• Function notation: is read as "f of x" or "f at 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 for x into the function's formula.

Example: If , then .

Graphs of Functions

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

• To graph a function, plot points for several values of x and connect them smoothly.

Example: To graph , plot points for x = -2, -1, 0, 1, 2, and connect them.

Graphing Functions by Plotting Points

• Select integer values for x within a reasonable range.

• Calculate the corresponding y-values using the function's formula.

• Plot the points (x, y) and connect them to reveal the graph's shape.

Example: For , plot points for x = -2, -1, 0, 1, 2 to obtain the parabola.

Vertical Line Test for Functions

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

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

Obtaining Information from a Function's Graph

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

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

Example: If the graph passes through (2, 5), then .

Identifying Domain and Range from a Function’s Graph

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

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

Example: If a graph extends from x = -3 to x = 4, the domain is [-3, 4]. If the lowest point is y = 0 and the highest is y = 5, the range is [0, 5].

Identifying Intercepts from a Function’s Graph

• x-intercepts: Points where the graph crosses the x-axis (set y = 0).

• y-intercept: Point where the graph crosses the y-axis (set x = 0).

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

Example: For , the x-intercepts are at x = -2 and x = 2, and the y-intercept is at y = -4.