Problem 59 - Beginning and Intermediate Algebra

Question

What is the domain of a linear function?

Step-by-Step Solution

Answer

The domain of a linear function is all real numbers, denoted as \((-\infty, +\infty)\) or \(\{x \in \mathbb{R}\}\), as there are no restrictions on the input values for a linear function.

Step 1: Definition of a Linear Function

A linear function is a function of the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. It represents a straight line on the Cartesian plane when it is graphed. The slope \(m\) determines the direction and steepness of the line, while the y-intercept \(b\) determines the point where the line crosses the y-axis.

Step 2: Domain of a Function

The domain of a function is the set of all possible input values, or in other words, the set of all x-values for which the function is defined.

Step 3: Domain of a Linear Function

In the case of a linear function, the function is defined for all possible x-values. There are no restrictions or limitations on the input values for a linear function. The graph of a linear function extends indefinitely in both directions along the x-axis. Therefore, the domain of a linear function is all real numbers, which can be denoted as \((-\infty, +\infty)\) or using the set notation \(\{x \in \mathbb{R}\}\).