Functions and Their Graphs

The Graph of a Function

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

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

• Graph of a Function: The set of points (x, y) in the xy-plane that satisfy the function's equation.

Vertical-Line Test

The Vertical-Line Test is a method used to determine whether a graph represents a function. If every vertical line intersects the graph at most once, the graph is a function.

• Theorem: A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point.

• Application: This test helps distinguish functions from non-functions visually.

Examples: Identifying the Graph of a Function

Consider the following graphs:

• Example 1: The graph below passes the vertical-line test, so it represents a function.

• Example 2: The graph below fails the vertical-line test, as some vertical lines intersect the graph at more than one point. Therefore, it does not represent a function.

• Example 3: The graph below also fails the vertical-line test, so it does not represent a function.

Obtaining Information from the Graph of a Function

Once a graph is confirmed to represent a function, various properties and values can be extracted:

• Function Value: If (x, y) is a point on the graph of f, then y = f(x).

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

• Range: The set of all y-values that the function attains.

• Intercepts: Points where the graph crosses the axes (x-intercepts and y-intercepts).

• Intersection with Lines: The number of times a given line (such as y = c) intersects the graph can provide information about the function's behavior.

Example: Analyzing a Function's Graph

Suppose the graph of a function f is given. You can answer questions such as:

• What is f(a) for a specific value of a?

• What is the domain and range of f?

• Where are the intercepts?

• For which x-values does f(x) = c?

• For which x-values is f(x) > 0 or f(x) < 0?

Application Example: Cost Function for a Flight

Consider a real-world function: The cost C per passenger for a trans-Atlantic flight depends on the ground speed x. The function might be given by an equation such as:

• Cost Function: (where k is a constant representing fixed costs)

• Domain: The set of possible ground speeds (e.g., x > 0)

• Table of Values: To analyze the function, create a table showing C(x) for various values of x.

Minimizing Cost: The ground speed that minimizes C(x) can be found by analyzing the graph or table.

Additional info: The actual formula and values for C(x) are not provided in the original material, so the above is a logical academic expansion based on typical cost functions in precalculus applications.