Problem 49 - College Algebra

Question

The domain of each piecewise function is \((-\infty, \infty)\) 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 range of the function is {3, -3}.

Step 1: Identify the conditions for the Piecewise Function

The function \(f(x)\) defined is a piecewise function with two conditions which can be divided as two individual problems. The first condition of the function states that \(f(x) = 3\) if \(x \leq -1\), and the other part of the function states that \(f(x) = -3\) if \(x > -1\).

Step 2: Graphing the function considering each condition

For \(x \leq -1\), the graph is a horizontal line passing through the point \((0, 3)\). It includes all the points to the left of \(x = -1\), including \(-1\). This line stops at \(x = -1\) and does not extend further because the line \(f(x) = -3\) begins from \(x > -1\). Hence graph the line from \(x = -1\) including \(-1\) to \((- \infty , 3)\). For \(x > -1\), the graph is another horizontal line passing through the point \((0, -3)\). This line starts at \(x = -1\), but does not include \(-1\), and extends to the right till \(x = \infty\). Thus the line reaches from \((-1, -3)\) to \((\infty , -3)\), excluding \(-1\).

Step 3: Determine the function's range

Once the graph is constructed, the range can be observed. The range of a function consists of all possible values of the function. In this case, the function only takes on two values, namely 3 and -3, therefore, the range is {3, -3}.

Key Concepts

Domain and Range of Piecewise Functions

In the context of piecewise functions, the domain refers to all possible input values (typically represented by x) that can be plugged into the function. The range, on the other hand, refers to all possible output values (or y-values) that the function can produce. For the given piecewise function:

• For the part where \( f(x) = 3 \), the function is defined for all x-values less than or equal to -1. This means that the domain for this section is \( x \leq -1 \).
• For the part where \( f(x) = -3 \), the function is defined for x-values greater than -1, which forms the domain \( x > -1 \).

The overall domain of the piecewise function is the combination of these, resulting in \(( -\infty, \infty )\), which indicates the function is defined for all real numbers. For the range, which consists of all possible y-values, we observe from the graph that the function only outputs two specific values: 3 and -3, corresponding to each piece. Thus, the range of the function is \({3, -3}\). This reflects the constant output levels based on the given intervals within the domain.

Graphing Piecewise Functions

Graphing piecewise functions involves plotting each segment based on its specific conditions. For a function broken into sections, like our piecewise example, this means drawing different parts of the graph adhering to the given rules:

• For \( x \leq -1 \), the function outputs a constant value of 3, represented as a horizontal line across these x-values. This line extends to the left from \( x = -1 \) towards negative infinity, including the point at \( x = -1 \).
• For \( x > -1 \), the function shifts to output a constant value of -3. A new horizontal line is drawn starting just to the right of \( x = -1 \) and extends infinitely to the right, covering all greater x-values.

When graphically representing these segments, it is important to use solid lines to indicate included points (such as \( x = -1 \) for the upper segment), and hollow or open indicators for points not included (like the lower segment starting just after \( x = -1 \)). This differentiation helps viewers understand where exactly the function switches its output behavior.

Understanding Horizontal Lines in Functions

Horizontal lines in the graph of a function indicate constant output across a range of x-values. In piecewise functions, these lines highlight sections of the domain over which the function's output remains unchanging. For our example:

• When \( f(x) = 3 \), the horizontal line sits at y = 3 across points where \( x \leq -1 \). This indicates for every x less than or equal to -1, the function returns the same output, 3.
• For \( f(x) = -3 \), the horizontal line at y = -3 stretches across the points where \( x > -1 \), delivering a consistent output of -3 regardless of x-values in that interval.

Horizontal lines are a visual cue in graphs that signify stability in function output. Because each segment represents a different condition of the piecewise function, horizontal lines efficiently communicate that within each section, the value does not fluctuate with changes in x. This is particularly important for interpreting each part of the piecewise function and understanding how these different constant sections fit together to form a complete function.