Problem 27 - Algebra and Trigonometry

Question

A function \(f\) is given. (a) Use a graphing calculator to draw the graph of \(f .\) (b) Find the domain and range of \(f\) from the graph. $$ f(x)=x-1 $$

Step-by-Step Solution

Answer

Domain: \((- fty, fty)\); Range: \((- fty, fty)\).

Step 1: Analyze the Function

The given function is a linear function of the form \(f(x) = x - 1\). This means the graph of the function will be a straight line with a slope of 1 and a y-intercept at \(-1\).

Step 2: Plot the Graph

Using a graphing calculator or plotting software, plot the function \(f(x) = x - 1\). The graph should be a straight line that intersects the y-axis at \(-1\) and has an upward slope.

Step 3: Determine the Domain

For linear functions like \(f(x) = x - 1\), there are no restrictions on the values that \(x\) can take. Therefore, the domain of \(f\) is \(\text{all real numbers}\), denoted as \((-fty, fty)\).

Step 4: Determine the Range

The range of a linear function like \(f(x) = x - 1\) covers all possible output values as \(x\) varies over all real numbers. Hence, the range is also \(\text{all real numbers}\), denoted as \((-fty, fty)\).

Key Concepts

Graphing Linear Equations

Understanding how to graph linear equations is an essential skill in algebra. A linear equation can be expressed in the form of \(y = mx + b\), where \(m\) denotes the slope and \(b\) represents the y-intercept. When graphing the equation \(f(x) = x - 1\), you'll start by locating the y-intercept on a coordinate plane. In this instance, the y-intercept is at \(-1\) on the y-axis. From this point, utilize the slope to determine the direction and steepness of the line. A slope of 1 means that for every unit you move to the right along the x-axis, you'll move up one unit along the y-axis. This ensures the line rises diagonally from left to right, maintaining a consistent upward motion. Using graphing tools or graph paper can aid in plotting this line precisely, transforming the abstract function into a visual representation.

Domain and Range

The concepts of domain and range offer insight into what values a function can accept (domain) and produce (range). For the function \(f(x) = x - 1\), determining these sets is straightforward.

• The domain includes all possible x-values you might input into the function. Linear functions do not have breaks or holes in their graphs, which means the domain for \(f(x) = x - 1\) spans all real numbers, denoted as \((-\infty, \infty)\).
• The range, on the other hand, represents all potential y-values that the function can output. Given the unrestricted nature of linear functions—without peaks or troughs—the range also encompasses all real numbers, \((-\infty, \infty)\).

Understanding domain and range is key to grasping how functions behave over different inputs and to predicting the potential outcomes of those inputs.

Slope-Intercept Form

The slope-intercept form is a clever way to express linear equations, which helps easily identify key graph features. Written as \(y = mx + b\), it highlights two main components: the slope \(m\) and the y-intercept \(b\).
The slope of a line is a measure of its steepness or incline. In mathematical terms, the slope \(m\) is the ratio of the change in y to the change in x, often described as "rise over run." In the case of the equation \(f(x) = x - 1\), the slope is 1, meaning the line inclines at a consistent angle of 45 degrees, moving one unit up for every unit it moves right.
The y-intercept \(b\) is where the line crosses the y-axis. Here, \(b = -1\), indicating the line intersects the y-axis one unit below the origin.
Employing the slope-intercept form not only simplifies the graphing process but also provides a clear understanding of how linear equations behave visually on a graph.