Chapter 2: Functions and Their Graphs

Section 2.2: The Graph of a Function

This section explores how to identify the graph of a function and how to extract information from such graphs. Understanding these concepts is fundamental for analyzing mathematical relationships and interpreting real-world data using functions.

Identifying the Graph of a Function

The graph of a function is the set of points \((x, y)\) in the xy-plane that satisfy the function's equation. Not every collection of points represents a function. For a relation to be a function, each value of x in the domain must correspond to exactly one value of y in the range.

• Definition: A function is a relation in which each input (x-value) has exactly one output (y-value).

• Vertical-Line Test: A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph at most once.

• Key Point: If a vertical line crosses the graph more than once, the relation is not a function.

Example: Identifying Graphs of Functions

Consider the following graphs:

• Graph (a): This graph passes the vertical-line test, so it represents a function.

• Graph (b): This graph fails the vertical-line test, so it does not represent a function.

• Graph (c): This graph fails the vertical-line test, so it does not represent a function.

• Graph (d): This graph passes the vertical-line test, so it represents a function.

Example Images:

Graph (a): Function (passes vertical-line test)

Graph (b): Not a function (fails vertical-line test)

Graph (c): Not a function (fails vertical-line test)

Obtaining Information from or about the Graph of a Function

Once a graph is identified as a function, you can extract valuable information from it. The point \((x, y)\) on the graph means that y = f(x). Conversely, if f(x) = y, then \((x, y)\) is a point on the graph.

• Domain: The set of all possible x-values for which the function is defined.

• Range: The set of all possible y-values that the function can take.

• Intercepts: Points where the graph crosses the axes. The x-intercept is where y = 0, and the y-intercept is where x = 0.

• Example: If the point \((-1, 3)\) is on the graph of f, then f(-1) = 3.

Example: Analyzing a Function's Graph

Given a graph of a function f:

• Find values: Determine f(a) for specific values of a by locating the corresponding point on the graph.

• Domain and Range: Identify the domain and range by observing the extent of the graph along the x- and y-axes.

• Intercepts: List all points where the graph crosses the axes.

• Intersection with lines: Count how many times a given line (e.g., y = c) intersects the graph.

• Solving equations: Find values of x for which f(x) = k or f(x) > k.

Example: Cost Function for a Trans-Atlantic Flight

Suppose the cost C per passenger for a flight is given by a function of ground speed x:

• Formula:

• Application: Calculate C(x) for specific values of x (e.g., 470 mph).

• Domain: Identify the set of possible ground speeds for which the cost function is defined.

• Graphing: Use a graphing calculator to visualize C(x) over a range of x values.

• Table: Create a table of values for C(x) at intervals (e.g., 450, 500, 550, 600 mph).

• Optimization: Find the ground speed that minimizes the cost per passenger.

Additional info: Table values are inferred as placeholders; actual values depend on the specific cost function formula.

Summary Table: Vertical-Line Test for Functions