Problem 65 - Precalculus Essentials

Question

The domain of each piecewise function \(i s(-\infty, \infty)\) a. Graph each function. b. Use your graph to determine the function's range. $$f(x)=\left\\{\begin{array}{rll} 3 & \text { if } & x \leq-1 \\ -3 & \text { if } & x>-1 \end{array}\right.$$

Step-by-Step Solution

Answer

The graph of function \(f(x)\) is two constant lines y=3 and y=-3 split at x=-1. The range of function \(f(x)\) is {3, -3}.

Step 1: Understanding the Piecewise function

The function \(f(x)\) defined in the problem is a piecewise function. It splits the x-axis into two regions and defines the function differently in each region. For \(x \leq -1\), \(f(x) = 3\), which means that no matter at what value the x is at or below -1, the function will always output '3'. Similarly, for \(x > -1\), \(f(x) = -3\), implying that for all x values greater than -1, the function's value will be -3. Therefore, the function will appear as two constant lines y=3 and y=-3, split at x=-1.

Step 2: Graphing the function

To graph this function, imagine two constant line graphs: one for \(f(x) = 3\) and the other for \(f(x) = -3\). However, we need to make a distinction at x=-1 according to the definition of our piecewise function. For \(x \leq -1\), i.e., to the left of -1 inclusively, we graph the line \(f(x) = 3\), and for \(x > -1\), i.e., to the right of -1, the line will be \(f(x) = -3\).

Step 3: Determining the Range

The range of a function is the set of all possible outputs or y-values that we can get from plugging in all possible x-values from the domain. Looking at our graph, we can clearly see that there are only two possible y-values obtainable from this graph irrespective of the x-values. These values are 3 and -3. Therefore, in this case, the range of \(f(x)\) is {3, -3}.

Key Concepts

Graphing Piecewise Functions

Understanding how to graph piecewise functions is essential for visualizing the behavior of a function defined by different rules over different intervals. With piecewise functions like the one in our example, \( f(x) \), it's important to graph each piece according to its own rule and the specific interval it applies to.

To graph \( f(x) \) efficiently, start by identifying the critical points where the function's rule changes—in this case, at \( x = -1 \). Next, draw the first segment. For \( x \leq -1 \) on the x-axis, sketch a horizontal line at \( y = 3 \). This represents that \( f(x) = 3 \) in this interval. It's helpful to use an open or closed circle to indicate whether the endpoint is included, which it is here, so we use a closed circle at \( x = -1 \) on the line \( y = 3 \).

For the second piece, draw a horizontal line at \( y = -3 \) for \( x > -1 \) on the x-axis. Because \( x = -1 \) is not included in this interval (the problem says 'greater than', not 'greater than or equal to'), use an open circle at \( x = -1 \) on this line. With these guidelines, graphing piecewise functions can be made straightforward and clear.

Domain and Range of Functions

The domain and range of a function are foundational concepts that tell us about the inputs a function can accept (domain) and the outputs it can produce (range).

The domain of our given function \( f(x) \) is \( (-\infty, \infty) \), meaning that \( x \) can be any real number. Visually, this represents a horizontal extent along the x-axis that spans from left to right indefinitely.

Exploring the Range

To determine the range, examine the y-values that correspond to these x-values after applying the function's rules. In our piecewise function, there are only two y-values produced, \( 3 \) and \( -3 \) as the output is fixed regardless of the input within each defined interval. Therefore, unlike the domain, which is continuous, the range is discrete and consists of just these two values: \( \{3, -3\} \). Understanding the difference between domain and range—and the way in which they relate—helps students analyze any function more effectively.

Step-By-Step Function Analysis

Breaking down the function analysis into manageable steps ensures that each property and characteristic is carefully considered. For a piecewise function, these steps help dissect the varying behavior depending on the input values.

Begin with identifying the intervals and corresponding function rules as shown in the function definition. Analyze each piece independently, looking at its slope (in the case of linear functions), y-intercept, and whether it includes the boundary point. By doing so, we can notice patterns and features of each piece that contribute to the function as a whole.

Verifying continuity at the boundary points where the rules change is another crucial step. In our function, there is a clear jump discontinuity at \( x = -1 \) since the two function pieces do not connect, which is a trait of this piecewise function. Lastly, examining the function for symmetry, potential asymptotes, or any behavior as \( x \) approaches \( \pm\infty \) rounds out the analysis, providing a comprehensive understanding of the function. This step-by-step approach streamlines the analytical process, leading to better comprehension and the ability to tackle a variety of function types.