Problem 35 - Algebra and Trigonometry

Question

Sketch the graph of the piecewise defined function. \(f(x)=\left\\{\begin{array}{ll}{3} & {\text { if } x<2} \\ {x-1} & {\text { if } x \geq 2}\end{array}\right.\)

Step-by-Step Solution

Answer

A horizontal line at \( y = 3 \) for \( x < 2 \) and a line \( y = x - 1 \) starting at \( x = 2 \).

Step 1: Understand the Piecewise Function

The piecewise function given is defined in two parts: 1. For any value of \( x \) where \( x < 2 \), the function \( f(x) \) is a constant value of 3. 2. For any value of \( x \) where \( x \geq 2 \), the function is defined by the expression \( f(x) = x - 1 \). We need to graph both parts separately to understand the overall behavior of \( f(x) \).

Step 2: Graph for \( x < 2 \)

Since \( f(x) = 3 \) for \( x < 2 \), this portion of the graph will be a horizontal line at \( y = 3 \) extending up to but not including the point where \( x = 2 \). At \( x = 2 \), there will be an open circle on the line to indicate that \( x = 2 \) is not included in this part of the function.

Step 3: Graph for \( x \geq 2 \)

For \( x \geq 2 \), \( f(x) = x - 1 \). This describes a linear function with a slope of 1 and a y-intercept of -1. We start plotting this from \( x = 2 \) using a filled dot at the point (2, 1), because when \( x = 2 \), \( f(2) = 2 - 1 = 1 \). Continue this line across any greater \( x \) values to visualize the line extending towards infinity.

Step 4: Combine Both Parts of the Graph

Combine the horizontal line from Step 2 and the linear line from Step 3 to form the complete piecewise graph. Make sure that there is an open circle at the endpoint of the first line at \( (2, 3) \) and a filled circle at the start of the linear line at \( (2, 1) \).

Step 5: Review the Graph

Check to ensure that each part of the piecewise function is represented accurately. The graph should show a horizontal segment at \( y = 3 \) for \( x < 2 \) and an inclined line starting at \( y = 1 \) for \( x = 2 \) and beyond, indicating the transition at \( x = 2 \).

Key Concepts

Graphing Functions

When graphing piecewise functions, it's important to clearly understand the behavior of each segment. In our example, the function changes its rule at the point where \( x = 2 \). Before this point, the function outputs a constant value, and after this point, it transitions to a different formula. Here’s a step-by-step guide:

• Identify the segments of the function. For \( f(x) \), there are distinct parts: for \( x < 2 \), and for \( x \geq 2 \).
• Graph each part separately, switching from one rule to the next at the transition points.
• Ensure accuracy with open or closed circles at transition points. Open circles indicate that a point is not included, while filled circles indicate inclusion.

Graphing these functions visually communicates their mathematical behavior, helping you to see how the function behaves over different ranges of \( x \). Break the task down into these steps to make graphing more manageable and accurate.

Linear Functions

Linear functions describe straight lines, characterized by the equation \( y = mx + b \). In our piecewise function, the portion defined by \( f(x) = x - 1 \) forms a linear function. Here's a breakdown:

• Intercept: For \( f(x) = x - 1 \), the y-intercept is -1. This is the point where the line crosses the y-axis, though in our context, we start considering the line only from \( x = 2 \).
• Slope: The coefficient of \( x \) is 1, giving a slope of 1. This means for every unit increase in \( x \), \( f(x) \) increases by 1 unit. It indicates a 45-degree angle line sloping upwards.

To graph this segment, mark the starting point at \( x = 2 \) with a filled circle at \( (2, 1) \), as this is where this linear rule starts applying. Then, extend the line upwards and to the right, recognizing how the slope determines its incline.

Function Behavior

Understanding function behavior involves examining how different parts react under varying inputs. A piecewise function, like our example, allows different rules to apply to different intervals of \( x \):

• Stability: For \( x < 2 \), the function remains stable at 3, showing horizontal behavior.
• Change: At \( x = 2 \), the behavior shifts to a changing function (\( f(x) = x - 1 \)), introducing a growth pattern.

Such shifts in function behavior can represent different phenomena in real-world scenarios, such as cost structures that vary with quantity or changes in speed over time. When interpreting these functions, look for critical transition points, like \( x = 2 \) in our case, and observe how these influence overall behavior. Recognizing these can enhance your understanding of mathematical modeling and its practical applications.