Problem 33 - Algebra A Combined Function

Question

Graph each piecewise-defined function. $$ f(x)=\left\\{\begin{array}{ll} 5 & \text { if } \quad x<-2 \\ 3 & \text { if } \quad x \geq-2 \end{array}\right. $$

Step-by-Step Solution

Answer

Graph two horizontal lines: one at \(y = 5\) for \(x < -2\) with an open circle at \(( -2, 5)\), and one at \(y = 3\) for \(x \geq -2\) with a filled circle at \(( -2, 3)\).

Step 1: Understand the Function Definition

The function given is a piecewise function, which means it is defined by different expressions over different intervals of the domain. Here, the function is split into two parts. For \(x < -2\), \(f(x) = 5\), and for \(x \geq -2\), \(f(x) = 3\).

Step 2: Identify the Intervals and Values

Identify the intervals: For \(x < -2\), \(f(x) = 5\), which is a constant function. For \(x \geq -2\), \(f(x) = 3\), which is also a constant function.

Step 3: Plot the Points for Each Interval

Plot the points for the interval \(x < -2\). Any point like \((-3, 5), (-4, 5)\) etc., but make it clear that it does not reach -2. At \(x = -2\), there is an open circle at \(( -2, 5)\) indicating that -2 is not included in this interval.

Step 4: Plot the Points for the Second Interval

Plot the points for the interval \(x \geq -2\). This will include points like \((-2, 3), (0, 3)\) and continues indefinitely to the right. At \(x = -2\), use a filled circle at \(( -2, 3)\) indicating that -2 is included.

Step 5: Complete the Graph

Combine the points to visualize the graph: for \(x < -2\), draw a horizontal line at \(y = 5\) with an open circle at \(( -2, 5)\). For \(x \geq -2\), draw another horizontal line at \(y = 3\) starting with a filled circle at \(( -2, 3)\).

Key Concepts

Constant Functions

A constant function is a type of function where the output (or y-value) remains the same, regardless of the input (or x-value). This means that no matter what value of x you choose, the output y is a fixed, constant number.
Think of it as a flat line on a graph. For example, if our function is defined as \(f(x) = 5\), it will produce a horizontal line at y = 5, regardless of the x-values you choose.
In the piecewise function \\(f(x)=\left\{\begin{array}{ll}\ 5 & \text { if } \quad x<-2 \ 3 & \text { if } \quad x \geq-2 \end{array}\right.\),\ both segments, \(f(x) = 5\) and \(f(x) = 3\), are constant functions.

• For \(x < -2\), the function stands at 5, thus the constant y-value is 5.
• For \(x \geq -2\), the function takes the value 3, so the constant y-value is 3.

Constant functions reveal their nature clearly through the simplicity of their horizontal line graphs, which makes them easy to work with and understand.

Intervals

In mathematics, an interval is a range of numbers that include all numbers between a start point and an endpoint. In our exercise, the function is defined over two intervals:

• One interval is \(x < -2\), which represents all numbers less than -2.
• The other interval is \(x \geq -2\), which includes all numbers greater than or equal to -2.

For each interval, the function adopts a different value, but within any specific interval, the function behaves consistently. This is a defining feature of piecewise functions, where different rules apply to different sections of the domain, creating distinct intervals. Understanding intervals helps us know when and how the rules change in a piecewise function.

Graphing

Graphing a piecewise function involves plotting each part of the function's specific rule separately on the same set of axes. For the given function, \(f(x)\) has two segments based on the intervals of x:

• For \(x < -2\), plot points where \(y = 5\) to create a horizontal line. Since \(-2\) is not included in this interval, use an open circle at \(( -2, 5)\) to represent that \(-2\) is not part of this segment.
• For \(x \geq -2\), plot points where \(y = 3\) creating another horizontal line starting at \(( -2, 3)\). Use a filled circle here to indicate inclusion of \(-2\) in this interval.

By graphing each interval separately, then connecting them without altering each distinct rule, you can accurately depict the piecewise function. This method allows for a clear visual understanding of how the function behaves over its entire domain.

Domain

The domain of a function is the complete set of possible input values (x-values), which are allowed by the function’s definition. For the given piecewise-defined function, different intervals of the domain are specified:

• For \(f(x) = 5\), the domain is \(x < -2\). This means \(x\) can be any number less than \(-2\).
• For \(f(x) = 3\), the domain is \(x \geq -2\). This allows \(x\) to be any number greater than or equal to \(-2\), including \(-2\) itself.

Understanding the domain is crucial, as it defines the scope within which each part of the piecewise function is applicable. By clearly understanding the domain, you are better equipped to work with and graph piecewise functions, as it determines where each segment starts and stops.