Problem 40 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph consists of horizontal lines for \( x <-1 \) and \( x > 1 \), and a diagonal line for \(-1 \leq x \leq 1\).

Step 1: Understand the piecewise function

The given piecewise function is defined in three parts: 1. For \( x < -1 \), the function \( f(x) = -1 \).2. For \( -1 \leq x \leq 1 \), the function \( f(x) = x \) (a linear function).3. For \( x > 1 \), the function \( f(x) = 1 \).Each segment has its own rule for determining the value of \( f(x) \), based on the value of \( x \).

Step 2: Sketch the first segment

Consider the segment where \( x < -1 \). For these values of \( x \), \( f(x) = -1 \), which means the graph will be a horizontal line at \( y = -1 \) extending leftwards from \( x = -1 \). Include an open circle at \( (x, y) = (-1, -1) \) because it is not included in this segment.

Step 3: Sketch the second segment

Now, focus on the interval \( -1 \leq x \leq 1 \). Here, \( f(x) = x \), forming a diagonal line with a slope of 1, passing through the points \((-1, -1)\) and \((1, 1)\). Since the endpoint \((-1, -1)\) is included, mark it with a closed circle; similarly, \( (1, 1) \) will also have a closed circle.

Step 4: Sketch the third segment

Finally, consider the part where \( x > 1 \). Here, \( f(x) = 1 \), resulting in a horizontal line at \( y = 1 \), extending rightwards from \( x = 1 \). Use an open circle at \( (1, 1) \) to denote that this point is not included in this segment.

Step 5: Analyze the complete graph

Check that all parts of the graph are correctly drawn. Each condition defines a specific range of \( x \) with its own function behavior: a horizontal line at \( y = -1 \) for \( x < -1 \), a diagonal line for \( -1 \leq x \leq 1 \), and another horizontal line at \( y = 1 \) for \( x > 1 \). Ensure the transitions at \( x = -1 \) and \( x = 1 \) are appropriately marked with open or closed circles.

Key Concepts

Graph Sketching

Graph sketching involves drawing a visual representation of a mathematical function, and it's a crucial skill in mathematics.
For piecewise functions, like the one we are looking at here, it's important to carefully consider each segment of the function separately.
The given function is divided into three segments, each with a different rule for calculating the value of the function based on the input, or _x_-value.

• The first segment, for values where \( x < -1 \), involves plotting a horizontal line at \( y = -1 \). Since this line does not include \( x = -1 \), it's essential to mark the endpoint with an open circle. This suggests that the point \( (-1, -1) \) is not part of this segment.


• Next, consider the interval \( -1 \leq x \leq 1 \), where the function \( f(x) = x \) is a diagonal line. Here, you’ll draw a line passing through points \( (-1, -1) \) and \( (1, 1) \), marking both points with closed circles to indicate they are included in this segment.


• The final segment, where \( x > 1 \), requires you to draw another horizontal line, this time at \( y = 1 \), starting from \( x = 1 \) with an open circle to indicate that \( (1, 1) \) is not included in this part of the function.

This approach ensures that each region of the piecewise function is accurately represented in the sketch.

Linear Functions

Linear functions are fundamental in mathematics, characterized by the equation \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
In our function, the linear part occurs when \(-1 \leq x \leq 1\) and is defined by \( f(x) = x \).
Here, the slope \( m \) is 1, meaning that for every increase of 1 unit in \( x \), \( f(x) \) also increases by 1 unit. When sketching linear functions, it's helpful to identify key characteristics:

• A diagonal line indicates that the relationship between \( x \) and \( y \) is proportional.
• The slope provides the direction and steepness of the line, with a positive slope resulting in an upward incline.
• The function passes through the origin or shifts depending on the y-intercept \( b \), but since \( b = 0 \) in this case, the line runs directly through the origin.

Understanding linear functions is pivotal for constructing graphs accurately as they often represent dynamic quantities such as speed, cost, or distance in real-world scenarios.

Function Analysis

Function analysis is the process of understanding a function's behavior and characteristics.
With piecewise functions, this means interpreting each segment and transition uniquely, especially focusing on endpoints and continuity.For our piecewise function:

• It contains constant segments where \( f(x) = -1 \) and \( f(x) = 1 \), resulting in horizontal lines for \( x < -1 \) and \( x > 1 \) respectively. These segments have no slope and signify stability or equality in the quantity represented.
• The linear segment \( -1 \leq x \leq 1 \) is marked by a slope of 1, embodying linear growth. This indicates consistency in change and transition between these two _x_-values.
• Analyzing the transitions at \( x = -1 \) and \( x = 1 \) highlights the function’s discontinuities—places where the graph "jumps". Specifically, the open and closed circles denote undefined and defined points, essential for understanding where the function is valid.

By breaking down the function into these components, we can grasp not just how it looks but what it truly represents.