Functions and Their Representations

Definition of a Function

A function is a special type of relation in mathematics. It is defined as a relation in which each possible input value (domain) leads to exactly one output value (range). This property ensures that the function is predictable: knowing the input allows us to determine the output.

• Relation: A set of ordered pairs (x, y).

• Function: A relation where each input (x) has only one output (y).

• Non-function: A relation where at least one input is paired with more than one output.

Example: The set {(1, 2), (2, 3), (3, 4)} is a function, but {(1, 2), (1, 3)} is not.

Determining Functions from Graphs

Not every collection of points in the xy-plane represents a function. To determine if a graph represents a function, we use the vertical line test:

• 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 1:

Graph A passes the vertical line test; Graph B does not, since a vertical line can intersect the circle at two points.

Examples: Parabolas and Sideways Parabolas

Example 2:

• is a function because each x-value has only one y-value.

• is not a function of x, since some x-values correspond to two y-values (e.g., gives and ).

Determining Functions from Equations

To determine if an equation defines y as a function of x, solve for y and check if each x gives only one y.

Example 3:

Additional info: Since the equation involves a ±, for some x-values there are two possible y-values, so this is not a function.

Obtaining Information from Graphs and Tables

Reading Values from a Graph

Given the graph of a function , you can find:

• f(a): The value of the function at (read the y-value at ).

• x when : The x-value(s) where the function equals (find where the graph crosses ).

Example 4:

• Find : Locate on the x-axis and read the corresponding y-value on the graph.

• Find when : Find the x-value(s) where the graph crosses .

Reading Values from a Table

Tables can also represent functions by listing input-output pairs.

Example 5:

• To evaluate , find the value in the table where (here, ).

• To solve , find all x-values where (here, ).

Summary Table: Ways to Represent a Function