Introduction to Functions and Linear Functions

Functions and Their Graphs

Definition of a Function

A function is a relation that assigns to each element in a set called the domain exactly one element in a set called the range. Functions are fundamental objects in mathematics and are used to model relationships between quantities.

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

• Range: The set of all possible output values (usually f(x) or y-values).

• Notation: If f is a function, we write f: X → Y, where X is the domain and Y is the range.

Example: If f(x) = x + 2, and the domain is {1, 2, 3}, then the range is {3, 4, 5}.

Graph of a Function

The graph of a function is the set of all points (x, f(x)) in the coordinate plane. For a function to be valid, each x-value must correspond to exactly one y-value.

• The graph of a linear function is a straight line.

• Other common function graphs include parabolas (quadratic functions), cubic curves, and more.

Example: The graph of f(x) = x is a straight line passing through the origin with a slope of 1.

Types of Functions

Linear Functions

A linear function is a function of the form f(x) = mx + c, where m and c are constants. The graph of a linear function is always a straight line.

• Slope (m): Measures the steepness of the line.

• Y-intercept (c): The point where the line crosses the y-axis.

Example: For f(x) = 2x + 1, the slope is 2 and the y-intercept is 1.

Vertical Line Test

The vertical line test is a 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.

• If every vertical line crosses the graph at most once, it is a function.

• If a vertical line crosses more than once, it is not a function.

Example: The graph of x = 2 is a vertical line and does not represent a function.

Properties of Linear Functions

Slope of a Line

The slope of a line measures the rate of change of the function. It is calculated as the ratio of the change in y to the change in x between two points on the line.

• Given two points (x1, y1) and (x2, y2), the slope m is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• If m > 0, the line rises as x increases.

• If m < 0, the line falls as x increases.

• If m = 0, the line is horizontal.

• If the line is vertical, the slope is undefined.

Example: For points (1, 2) and (3, 6), the slope is $m = \frac{6-2}{3-1} = 2$.

Special Cases: Vertical and Horizontal Lines

Vertical and horizontal lines are special cases of linear equations:

• Vertical line: x = a (slope is undefined)

• Horizontal line: y = b (slope is 0)

Example: The line x = 2 is vertical; the line y = -3 is horizontal.

Summary Table: Linear Functions and Their Properties

Type of Line	Equation	Slope	Graph
General Linear	y = mx + c	m	Straight line
Vertical	x = a	Undefined	Vertical line
Horizontal	y = b	0	Horizontal line

Additional info: These notes cover the foundational concepts of functions, their graphical representation, and the properties of linear functions, which are essential for further study in Precalculus.