Functions: Domain, Range, and Function Notation

Functions and Their Properties

Relations, Domain, and Range

A relation is any set of ordered pairs, typically written as (x, y). The domain of a relation is the set of all first elements (x-values), while the range is the set of all second elements (y-values).

• Domain: All possible input values (x-values) for the relation.

• Range: All possible output values (y-values) for the relation.

Example: Given the relation { (1, 2), (3, 4), (5, 6) }, the domain is {1, 3, 5} and the range is {2, 4, 6}.

Functions

A function is a special type of relation in which each element of the domain corresponds to exactly one element in the range. In other words, for every x-value, there is only one y-value.

• Key Point: No x-value is paired with more than one y-value.

• Non-Example: The set { (2, 3), (2, 5) } is not a function because the input 2 is paired with two different outputs.

Vertical Line Test

The Vertical Line Test is a graphical method to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.

• How to Apply: Move a vertical line across the graph. If it ever touches the graph in more than one place at the same time, the graph is not a function.

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

Function Notation

Function notation is a way to name functions for clarity. The notation f(x) represents the value of the function f at the input x. It does not mean f times x.

• f(x): Read as "f of x"; represents the output when x is the input.

• y and f(x): Both represent the output value; f(x) is just a more specific notation.

• Example: If f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11.

Intercepts

• x-intercept(s): The point(s) where the graph crosses the x-axis. At these points, y = 0.

• y-intercept(s): The point(s) where the graph crosses the y-axis. At these points, x = 0.

Example: For the function f(x) = x^2 - 4:

• x-intercepts: Set f(x) = 0: $x^2 - 4 = 0$ $x^2 = 4$ $x = \pm 2$ So, the x-intercepts are (2, 0) and (-2, 0).

• y-intercept: Set x = 0: $f(0) = 0^2 - 4 = -4$ So, the y-intercept is (0, -4).

Interval Notation

Domains and ranges are often written in interval notation to describe all possible values.

• Example: If a function is defined for all real numbers x such that $-3 \leq x \leq 5$, the domain is written as [-3, 5].