Problem 15 - Precalculus

Question

In Exercises \(13-20\), sketch the graph of the given piecewise-defined function. $$ f(x)=\left\\{\begin{array}{rll} -3 & \text { if } & x<0 \\ 2 x-3 & \text { if } & 0 \leq x \leq 3 \\ 3 & \text { if } & x>3 \end{array}\right. $$

Step-by-Step Solution

Answer

The graph is made of three segments: a horizontal line at \\(y = -3\\), a line from (-3,0) to (3,3), and a horizontal line at \\(y = 3\\).

Step 1: Understanding the Function

The given function is a piecewise-defined function with three distinct rules. Each rule applies to a different domain of input values, which we must consider separately while sketching the graph.

Step 2: Analyzing the First Piece

For the first piece, \(f(x) = -3\), this applies to \(x < 0\). This implies that for any \(x\) less than 0, \(f(x)\) is constant at -3. Plot this as a horizontal line from \(-fordots\) up to just before \(x = 0\).

Step 3: Analyzing the Second Piece

For the second piece, \(f(x) = 2x - 3\), this rule applies for \(0 \leq x \leq 3\). To plot this section, calculate the value of \(f(x)\) at \(x = 0\): \(f(0) = 2(0) - 3 = -3\). Calculate \(f(3)\): \(f(3) = 2(3) - 3 = 3\). Plot these points and draw a line segment between them.

Step 4: Analyzing the Third Piece

For the third piece, \(f(x) = 3\), which applies to \(x > 3\), \(f(x)\) is constant at 3. Plot this as a horizontal line starting just after \(x = 3\) and extending towards positive infinity.

Step 5: Combining All Pieces

Combine the individual segments to form the complete piecewise graph. You should see a horizontal line at \(y = -3\) for \(x < 0\), a diagonal line from (0,-3) to (3,3), and then another horizontal line at \(y = 3\) for \(x > 3\). There should be a closed point at \(x = 0\) for the second segment and an open point at \(x = 3\) for the first segment.

Key Concepts

Graphing Functions

Graphing a piecewise function involves plotting each piece separately and ensuring a smooth transition between them. A piecewise function, like the one given, has separate equations for different intervals of the input variable, which means we'll graph several segments on the same set of axes.

• Identify the different pieces of the function: Each piece has its own specific rule for how to calculate the output values (like -3, or 2x - 3), and a domain (range of x-values) where that rule applies.
• Graph each segment separately: For the first piece, plot a horizontal line for all x-values less than 0. For the second piece, graph a straight line between x=0 and x=3. Finally, graph the third piece as a horizontal line for x-values greater than 3.
• Connect the pieces: Ensuring continuity, check each segment's endpoints to see if they should be open or closed dots, helping show whether the points at the boundaries are included or not.

This method of graphing lets you visually display how the function behaves differently across its domain. It turns complex mathematical expressions into understandable visual forms.

Domain and Range

In mathematics, understanding the domain and range of a piecewise function is crucial for correct graphing and analysis. The domain represents all the possible input values (x-values) that the function can accept. For a piecewise function, you must examine the domain for each segment separately.

• The first piece, where the function is constant at -3, is defined for all x-values less than 0.
• The linear piece of the function, defined as 2x - 3, spans the interval from 0 to 3 inclusive.
• The final constant segment, where the function equals 3, covers x-values greater than 3.

When considering the range, look at each segment:

• The first segment has a constant output of -3 for all x-values in its domain.
• The middle segment outputs values between -3 and 3 as x ranges from 0 to 3.
• The last segment gives a constant output of 3 for x-values over 3.

Understanding these intervals helps you identify how the output of a piecewise function (the range) relates to its inputs (the domain).

Function Analysis

Analyzing a piecewise function involves looking at each segment to understand what it says about the function's behavior. This analysis offers insights into how the function changes across its domain.

• Constant segments, like the first and third pieces, illustrate stability: regardless of the input within their respective domains, the output remains unchanged.
• The middle segment, a linear equation, displays a varying relationship between input and output. This means the output changes at a consistent rate as x moves from 0 to 3.
• Note transitions: At each boundary point (x=0, x=3) where pieces meet, consider if the function remains continuous. At x=0 the function continues smoothly, but at x=3, there's a jump from 3 back to another constant value of 3, displaying a unique feature of piecewise functions.

This type of analysis allows students to understand both the piecewise function’s overall behavior and how each individual piece contributes to that behavior. It also demonstrates how abrupt or gradual changes occur within different parts of the function's domain.