Functions and Their Graphs

Relations and Ordered Pairs

A relation in mathematics is a correspondence between two variables, typically expressed as an ordered pair (x, y). The first variable, x, is the independent variable, and the second, y, is the dependent variable. The set of all possible values for x is called the domain, while the set of all possible values for y is called the range.

• Domain: The set of input values (x-values).

• Range: The set of output values (y-values).

• Example: For , both domain and range are all real numbers ().

• Example: For , domain is all real numbers (), but range is since cannot be negative.

Definition of a Function

A function is a special type of relation where each element of the domain is paired with exactly one element of the range. In other words, for every input value (x), there is only one output value (y).

• Function Notation: denotes a function named 'f' with independent variable x. For example, .

• Criteria: An equation is a function if, for any x in the domain, there is only one corresponding y.

• Example: is a function because each x yields only one y.

• Non-Example: is not a function of x because for some x, there are two possible y values (e.g., gives and ).

Function Notation and Evaluation

Function notation provides a concise way to express functions and evaluate them for specific values.

• Notation: , , etc., where the letter inside the parentheses is the independent variable.

• Evaluation: Substitute the given value into the function and solve.

• Example: If , then .

• Example: If , then .

• Example with variable substitution: .

Determining if a Relation is a Function

To determine if a set of ordered pairs or an equation represents a function, check if each input value corresponds to only one output value.

• Ordered Pairs:

  • Example: is a function (each x has one y).

  • Non-Example: is not a function (x = 5 has two y values).

• Equations:

  • Example: is a function of x (for each x, only one y).

  • Non-Example: is not a function of x (for x = 1, y = 2 or y = -2).

Graphing Functions

Functions are graphed by plotting points corresponding to input-output pairs and connecting them. Linear functions require only two points, while higher-order functions may require more.

• Linear Function Example:

  • If ,

  • If ,

• Graphing Steps:

  1. Select values for x from the domain.

  2. Calculate corresponding y values.

  3. Plot the points (x, y) on the coordinate plane.

  4. Connect the points to reveal the graph.

• Example: , ; is 4 units below for each x.

Vertical Line Test

The vertical line test is a graphical method to determine if a graph represents a function. If any vertical line crosses the graph more than once, the graph is not a function.

• Example: Graphs of lines (except vertical lines) pass the vertical line test and are functions.

• Non-Example: A vertical line at fails the test; for , there are infinite y values.

Evaluating Functions for Given Values

To evaluate a function for a specific value, substitute the value into the function and solve.

• Example:

• Example:

• Example:

Finding Information from Graphs

Graphs can be used to determine function values, domain, range, and intercepts.

• Function Value: To find , locate on the x-axis and read the corresponding y value from the graph.

• Example: If , find and read .

• Intercepts:

  • x-intercept: Where the graph crosses the x-axis ().

  • y-intercept: Where the graph crosses the y-axis ().

Domain, Range, and Intercepts from Graphs

For a given graph, you can determine the domain, range, and intercepts as follows:

Summary Table: Function vs. Not a Function

Additional info:

• For higher-order functions (quadratic, absolute value, cubic), more points are needed to accurately graph the function.

• Intercepts are important for understanding the behavior of functions and their graphs.

• Function notation is used throughout mathematics to clarify the relationship between variables.