Problem 64 - Beginning and Intermediate Algebra

Question

Graph the following piecewise functions. $$f(x)=\left\\{\begin{array}{cc}2 x+13, & x \leq-4 \\\\-\frac{1}{2} x+1, & x>-4\end{array}\right.$$

Step-by-Step Solution

Answer

To graph the piecewise function, first graph the two functions individually for their respective intervals: \(f(x) = 2x + 13\) for \(x \leq -4\), and \(f(x) = -\frac{1}{2}x + 1\) for \(x > -4\). Then, combine the graphs on the same coordinate plane by removing any parts of the graphs not in the correct interval. For our example, there will be an abrupt break between intervals, as the two functions do not intersect at \(x = -4\).

Step 1: Identify the Functions and Intervals

The given piecewise function is: \[f(x) = \left\{ \begin{array}{cc} 2x + 13, & x \leq -4 \\ -\frac{1}{2}x + 1, & x > -4 \end{array} \right. \] The intervals and corresponding functions are: - \(x \leq -4\) with function \(f(x) = 2x + 13\) - \(x > -4\) with function \(f(x) = -\frac{1}{2}x + 1\)

Step 2: Graph the Functions Individually

We will graph each function separately and then merge them to create the piecewise graph. Function 1: \(f(x) = 2x + 13\) for \(x \leq -4\) 1. Start by creating a table with values for x and their corresponding y-values. 2. Choose some x-values less than or equal to -4 3. Graph these points and connect them with a line. Function 2: \(f(x) = -\frac{1}{2}x + 1\) for \(x > -4\) 1. Start by creating a table with values for x and their corresponding y-values. 2. Choose some x-values greater than -4 3. Graph these points and connect them with a line.

Step 3: Combine the Graphs

To create the graph of the piecewise function, we need to combine the graphs of the two individual functions on the same coordinate plane. Follow these steps: 1. Place the graph of \(f(x) = 2x + 13\) on the same graph as \(f(x) = -\frac{1}{2}x + 1\). 2. Remove any parts of the graphs that are not in the correct interval. Since we are only looking at the function \(f(x) = 2x + 13\) when \(x \leq -4\), we need to remove any parts of the graph where \(x > -4\). Similarly, for the function \(f(x) = -\frac{1}{2}x + 1\), remove any parts of the graph where \(x \leq -4\). 3. In some cases, the function may be continuous at the point where the intervals meet; if this is the case, we'd need to use an open circle and a closed circle to represent continuity/discontinuity between the intervals. However, in our example, the two functions do not intersect at \(x = -4\), so the graph will have an abrupt break between the intervals. At this point, you should have a complete graph of the piecewise function.

Key Concepts

Graphing Piecewise Functions

Graphs of piecewise functions are divided into distinct segments. Each segment represents a different part of the function defined over a particular interval. This means that piecewise functions can look a bit different compared to typical functions.
Rather than a smooth or continuous graph, a piecewise function might have breaks or shifts depending on its definition across its intervals.

In our exercise, we have a piecewise function with two separate rules. We need to graph each of these rules individually and then combine them to form the final graph.

• In piecewise function graphing, it's important to clearly mark the boundary between intervals. This is often done using open or closed circles. A closed circle means that a particular point is included in the interval. An open circle means it's not.
• Graph each piece individually over its designated interval whenever graphing piecewise functions.

If you follow these steps, you'll be able to accurately depict any piecewise function with ease.

Algebraic Expressions

Algebraic expressions form the basis for the equations seen in piecewise functions. Each segment of a piecewise function is an algebraic expression.
In our function, we have two different algebraic forms: a linear form \(2x + 13\) and another linear form \(-\frac{1}{2}x + 1\).

To understand the essence of these expressions, break them down:

• The first expression, \(2x + 13\), means every input \(x\) is multiplied by 2, with 13 added to the result.
• The second expression, \(-\frac{1}{2}x + 1\), means each input \(x\) is multiplied by \(-1/2\), then added to 1.

These expressions tell us how each segment behaves over its defined interval. Make sure you carefully identify these expressions and know how to calculate each corresponding \(y\) value by substituting \(x\) values in the intervals these expressions are defined for.

Function Intervals

Function intervals are crucial when working with piecewise functions. An interval is a domain segment over which a particular function expression holds true. In piecewise functions, these often dictate where one function stops, and another begins.
For the function to be accurate, know which algebraic expression applies to which interval.

For our specific piecewise function:

• The interval \(x \leq -4\) uses the function \(2x + 13\).
• The interval \(x > -4\) uses the function \(-\frac{1}{2}x + 1\).

Intervals guide which sections of the graph display which expressions. It's crucial to respect these limitations to ensure the graph accurately reflects the function's behavior over all sections.

Graph Interpretation

Interpreting the graph of a piecewise function involves understanding the transitions between the different function segments and how they are represented visually.
When viewing a completed piecewise graph:

• Notice the discontinuity or break at the intersection of the intervals. The graph will often have open and closed circles to indicate whether a point is included or excluded.
• Pay attention to the slopes of the different segments; these indicate how steeply the function changes over that particular interval.

You can interpret each section based on the underlying algebraic expression for that interval, and understand how they contribute to the function's overall behavior.
This can lead to greater insights, such as whether the functions are increasing or decreasing in each interval, and what happens as \(x\) approaches the boundary between intervals.