91Ó°ÊÓ

A linear function is one of the simplest and most fundamental types of functions in algebra. It is represented by the equation of the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. Here, \(m\) represents the slope of the line, while \(b\) is the y-intercept, the point where the line crosses the y-axis.
Linear functions graph as straight lines, making them easy to work with.This predictability makes understanding them crucial in algebra and real-world applications.
In our problem, the linear function is \(f(x) = 3x\). Here, the slope \(m\) is 3, and the y-intercept \(b\) is 0. This means the line crosses the origin \((0,0)\) and rises steeply because of the slope of 3.
The properties of linear functions are:

• Straight-line graph.
• Constant rate of change.
• The slope \(m\) determines the steepness and direction.
• Crosses the y-axis at \(b\).

The domain and range are essential concepts in understanding the behavior of functions.
The **domain** of a function is the set of all possible input values (usually represented as x) that the function can accept. For linear functions like \(f(x) = 3x\), you can plug in any real number for x. Thus, the domain is all real numbers, or written in interval notation as \((-fty, fty)\).
The **range** of a function is the set of all possible output values (usually represented as y). Since linear functions continue infinitely in both directions without hitting a maximum or minimum y-value, the range for \(f(x) = 3x\) is also all real numbers, written as \((-fty, fty)\).
To summarize for the function \(f(x) = 3x\):

• Domain: \((-fty, fty)\)
• Range: \((-fty, fty)\)

This means the function's graph extends infinitely in both the x-direction and the y-direction.

Graphing a function helps you visualize its behavior.
To graph the linear function \(f(x) = 3x\), you need to find and plot some key points. Here are the steps:

• **Choose values for x**: Start with simple values, for instance, \(x = 0\) and \(x = 1\).
• **Calculate y-values**: Plug the chosen x-values into the equation to find the corresponding y-values.
  For \(x = 0\): \(f(0) = 3(0) = 0\). This gives us the point (0,0).
  For \(x = 1\): \(f(1) = 3(1) = 3\). This gives us the point (1,3).
• **Plot the points**: Draw these points on a coordinate plane.
• **Draw the line**: Connect the points with a straight line.
  

Remember:

• The slope (3 in this case) tells you how steep the line is. Here, for every increase of 1 in x, y increases by 3.
• The y-intercept (if it’s not zero) shows where the line crosses the y-axis. Here, it crosses at (0,0).

Graphing is a crucial skill that helps in understanding the function’s behavior and its interaction with the axes.