Problem 62 - Beginning and Intermediate Algebra

Question

Graph the following piecewise functions. $$g(x)=\left\\{\begin{aligned}x-1, & x \geq 2 \\\\-3 x+3, & x<2\end{aligned}\right.$$

Step-by-Step Solution

Answer

To graph the given piecewise function, graph the function \(x-1\) for \(x \geq 2\) with a closed circle at point (2,1), then graph the function \(-3x+3\) for \(x<2\) with an open circle at point (2,-3). Combine the two graphs, making sure to respect the defined domains and the open/closed circles at the breaks between the functions.

Step 1: Graph the function \(x-1\) for \(x \geq 2\)

Identify the function \(x-1\) and its domain \(x \geq 2\). This is a linear function with a slope of \(1\) and a y-intercept at \(-1\). To graph this function, start at the y-intercept at point (0, -1) and draw a line with a slope of \(1\). Only include the part of the line where \(x \geq 2\). This means that the line should start at the point (2,1) and continue upwards and to the right. Use a closed circle (•) at the point (2,1) to indicate that the point is included in the graph.

Step 2: Graph the function \(-3x+3\) for \(x<2\)

Identify the function \(-3x+3\) and its domain \(x<2\). This is a linear function with a slope of \(-3\) and a y-intercept at \(3\). To graph this function, start at the y-intercept at point (0, 3) and draw a line with a slope of \(-3\). Only include the part of the line where \(x<2\). This means that the line should start at point (2, -3) and continue downwards and to the left. Use an open circle (○) at the point (2, -3) to indicate that the point is not included in the graph.

Step 3: Combine the graphs

Place the individual graphs of the functions \(x-1\) and \(-3x+3\) together on the same coordinate plane. Make sure to respect the defined domains and the open/closed circles at the breaks between the two functions. The final graph should show a line with a slope of \(1\) starting from point (2,1) and going upwards and to the right, and a line with a slope of \(-3\) starting from point (2, -3) and going downwards and to the left. The closed circle should be placed at point (2,1) and the open circle should be placed at point (2,-3).

Key Concepts

Graphing Linear Equations

Graphing linear equations is all about understanding and plotting lines on a coordinate plane.
A linear equation is typically in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Here's a simple guide to graphing these equations.

• **Slope (\(m\))**: This tells you how steep the line is. If the slope is positive, the line will go upwards from left to right. If negative, it will go downwards.
• **Y-intercept (\(b\))**: This is where the line crosses the y-axis. It's your starting point for drawing.

To graph a linear equation, start from the y-intercept and use the slope to determine the direction of the line. Remember, a slope of \(1\) means you go up one unit right for every unit you go over.
For piecewise functions, you'll apply this separately for each part of the function, like in the given exercise. Each piece could be graphed individually while considering its specific domain.

Domain and Range

The domain and range of a function describe the possible values of \(x\) and \(y\) that the function can take.
Understanding these concepts is crucial when graphing piecewise functions since each piece of the function has its own domain.

• **Domain**: This refers to all possible values of \(x\) for which the function is defined. When graphing, pay attention to whether the piece of the function is defined with a closed circle (point included) or an open circle (point not included).
• **Range**: This describes all possible values of \(y\) that the function can output. While less often directly plotted, it’s important for understanding the height and limits of the graph.

In piecewise functions, each segment only exists within its defined domain. Respecting the domain ensures that pieces of the function do not overlap incorrectly or extend where they shouldn't when you are plotting them.

Linear Functions

Linear functions are a simple yet powerful type of mathematical expression.
They are fundamental in understanding more complex concepts, like piecewise functions.
A linear function can be identified by a constant rate of change or slope.

• **Formula**: Linear functions are expressed as \(y = mx + b\). This clean formula makes it easy to predict and compute values.
• **Features**: Every linear function features a straight-line graph with a consistent slope. The x-intercept and y-intercept define specific points where the graph crosses the axes.
• **Example**: The function \(x-1\) has a slope of \(1\) and a y-intercept of \(-1\), while \(-3x + 3\) has a slope of \(-3\) and a y-intercept at \(3\).

Linear functions are easy to interpret and graph, making them an excellent starting point for studying more complicated functions composed of multiple linear segments, such as piecewise functions.