Linear Functions

A function is a bunch of ordered pairs of things (numbers), with the property that the first members of the pairs are all different from one another.

 {(1, 1), (2, 1), (3, 2)}

This function consists of three pairs, whose first members are 1, 2 and 3.
It is customary to give functions names, like f, g or h, and if we call this function f, we generally use the following notation to describe it:

f(1) = 1, f(2) = 1, f(3) = 2

The first members of the pairs are called arguments and the whole set of them is called the domain of the function. Thus the arguments of f here are 1, 2 and 3, and the set consisting of these three numbers is its domain.

The second members of the pairs are called the values of the functions, and the set of these is called the range of the function.

Example: If you stick a thermometer in your mouth, you can measure your temperature, at some particular time. You can define a function T or temperature, which assigns the temperature you measure to the time at which you remove the thermometer from your mouth. This is a typical function. Its arguments are times of measurement and its values are temperatures.

Linear functions are functions that have x as the input variable, and x is raised only to the first power. Such functions look like the ones in the above graphic. Notice that x is raised to the power of 1 in each equation.

Functions such as these yield graphs that are straight lines, and, thus, the name linear. Linear functions come in three main forms.

Slope - Intercept form:  Y= mx+c

• The graph of this equation is a straight line.
• The slope of the line is m.
• The line crosses the y-axis at b.
• The point where the line crosses the y-axis is called the y-intercept.
• The x, y coordinates for the y-intercept are (0, b).

Example: Graph: y = 2x + 1

Slope – Point form: y-y₁ = m (x – x₁)

• The graph of this equation is a straight line.
• A known point on the line is (x₁, y₁).
• The slope of the line is m.

Example: m= -3/4 starting at (-5, 3)

General form; Ax+By+C=0

• The graph of this equation is a straight line.
• The slope of the line is (-A/B).
• The y-intercept is (-C/B).

Example: 3x+2y=6