Problem 28 - Applied Calculus

Question

Graph each function. $$ f(x)=\left\\{\begin{array}{ll} 8-2 x & \text { if } x \geq 2 \\ x+2 & \text { if } x<2 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph the two linear pieces using their respective conditions: one from \(x \geq 2\) and the other from \(x < 2\).

Step 1: Understand the Piecewise Function

The function given is piecewise, meaning it has two separate expressions depending on the value of \(x\). For \(x \geq 2\), the function is \(f(x) = 8 - 2x\). For \(x < 2\), the function is \(f(x) = x + 2\). We'll graph each piece separately.

Step 2: Graph the Piece for \(x \geq 2\)

For the part of the function where \(x \geq 2\), use the equation \(f(x) = 8 - 2x\). This is a linear function with a slope of \(-2\) and a y-intercept at 8, but since it is only defined for \(x \geq 2\), plot points from \(x = 2\) onward, like (2, 4) and (3, 2). Use a solid dot at \(x = 2\) because \(x = 2\) is included in this piece.

Step 3: Graph the Piece for \(x < 2\)

For \(x < 2\), the equation \(f(x) = x + 2\) applies. This is also a linear function, with a slope of 1 and a y-intercept of 2. Plot points for \(x < 2\), such as (1, 3) and (0, 2). Use an open dot at \(x = 2\) because \(x = 2\) is not included in this piece.

Step 4: Combine Graphs on the Same Coordinate Plane

Combine the segments by plotting both on the same graph. Make sure to use the correct dot notation: solid for included values (like \(x \geq 2\) for the first piece) and open for not included values (like \(x = 2\) for the second piece). The graph will show two linear segments meeting at \(x = 2\), with no overlap or gap.

Key Concepts

Linear Functions

Linear functions are among the simplest and most important types of functions used in mathematics. They have the form \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.
This means that linear functions create straight lines when graphed. Their behavior is predictable and consistent, as they depict a constant rate of change.
The slope \( m \) indicates how steep the line is, and can be positive, negative, or zero, impacting the direction of the line.

• A positive slope means the line ascends as it moves from left to right.
• A negative slope results in the line descending.
• A slope of zero indicates a horizontal line.

Linear functions are fundamental to understanding more complex functions because they serve as building blocks, especially when dealing with piecewise functions, as each piece can be linear. When working with piecewise functions that involve linear components, it is crucial to manage where each linear expression applies based on the domain restrictions.

Graphing Techniques

Graphing piecewise functions involves a few specific techniques that help clearly illustrate how the different parts of the function behave. Essentially, you need to consider each piece separately and ensure they are accurately represented on the graph. Here are the basic steps:
Identify the different pieces of the function and the domain restriction for each segment.
For each piece, use the linear equation to find key points.
Plot these points on the coordinate plane, mindful of whether to use open or solid dots.

• Begin by plotting crucial points where the domain restrictions change the function's behavior, especially at the break point \( x \).
• For the domain \( x \geq 2 \), in the example, graph the line \( f(x) = 8 - 2x \). Use a solid dot where \( x = 2 \) because this point is included in the domain segment.
• For the domain \( x < 2 \), graph \( f(x) = x + 2 \) and place an open dot at \( x = 2 \). This signals that the function segment from \( x < 2 \) does not include this value.

Finally, combine the plots for all pieces to create the full graph, verifying there are no discontinuities (gaps or jumps) at the transition points, except where the piecewise definition requires them.

Domain and Range

Understanding the domain and range of a piecewise function is an essential part of analyzing how the function behaves and understanding its limitations. The domain refers to all the possible \( x \)-values for which the function is defined. In the given function, the domain is all real numbers since every real \( x \) value has a corresponding \( f(x) \) value.
The range, in contrast, consists of all possible \( y \)-values that the function can take. The range can vary depending on the different expressions in the piecewise function. For the function given:

• For the segment \( f(x) = 8 - 2x \) (for \( x \geq 2 \)), the range descends from 4 downward.
• For the segment \( f(x) = x + 2 \) (for \( x < 2 \)), the range starts from below 4 upward.

In analyzing the range, pay attention to the endpoints of the intervals, especially where the function's segments meet. For instance, while \( x = 2 \) is included in the first segment, it’s not in the second, which affects how you interpret the \( y \)-value continuity. Recognizing these differences helps in assessing the function's entire behavior visually and mathematically on the graph.