Problem 27 - Algebra 2

Question

Graph each function. Identify the domain and range. \(h(x)=\left\\{\begin{aligned}-1 & \text { if } x<-2 \\ 1 & \text { if } x>2 \end{aligned}\right.\)

Step-by-Step Solution

Answer

Domain: \(( -\infty, -2 ) \cup ( 2, \infty )\); Range: \{-1, 1\}.

Step 1: Analyze the Function

The function given is a piecewise function with two conditions. For values of \(x < -2\), the function equals \(-1\). For values of \(x > 2\), the function equals \(1\). Notice that the function has no definition for \(-2 \leq x \leq 2\).

Step 2: Determine the Domain

The domain of the function is the set of all permissible \(x\) values. Since the function is only defined for \(x < -2\) and \(x > 2\), the domain is \(( -\infty, -2 ) \cup ( 2, \infty )\).

Step 3: Determine the Range

The range is the set of all possible output values (\(h(x)\) values). From the given piecewise function, \(h(x) = -1\) for \(x < -2\) and \(h(x) = 1\) for \(x > 2\), meaning the range is \{-1, 1\}.

Step 4: Graph the Function

To graph the function, plot two horizontal lines. One line at \(y = -1\) corresponding to \(x < -2\). Then, draw another horizontal line at \(y = 1\) for \(x > 2\). Both lines do not include the points at \(x = -2\) and \(x = 2\), which should be indicated by open circles on the graph at these discontinuities.

Key Concepts

Domain

The domain of a function refers to all the possible input values (or "x values") that the function can accept. For a piecewise function like the one given, we need to carefully examine each piece to determine where the function is defined.

In our specific example, the function is defined by two conditions:

• For values of \(x < -2\), the function outputs \(-1\).
• For values of \(x > 2\), the function outputs \(1\).

There is a gap where the function does not have any outputs, specifically between \(-2\) and \(2\) inclusive. Thus, the domain excludes this interval.

As a result, the domain of the function is represented in interval notation as \(( -\infty, -2 ) \cup ( 2, \infty )\). This shows that the function accepts all real numbers below \(-2\) and above \(2\). Understanding the domain helps identify where the function is functioning and is critical for graphing accurately.

Range

The range of a function is all the possible output values the function can produce. In simpler terms, it's all the 'y-values' or the results of plugging input values into the function.

For our piecewise function, two distinct parts contribute to the range:

• When \(x < -2\), the function outputs \(-1\).
• When \(x > 2\), the function outputs \(1\).

Because these are the only outputs possible based on the given conditions, the range of the function is simply the set of these two values: \{-1, 1\}.

Thus, irrespective of how you might test different x-values within the domain, the result will always be either \(-1\) or \(1\). Grasping the range is important when predicting what kind of output can be expected from a function, and helps in verifying all possible function outcomes.

Graphing Functions

Graphing functions, especially piecewise functions, can initially seem complex, but breaking it down helps. For a piecewise function, each "piece"—or separate condition—must be graphed according to its defined interval.

In our function example, we need to graph two distinct horizontal lines:

• The first line is for \(x < -2\), at \(y = -1\).
• The second line is for \(x > 2\), at \(y = 1\).

To indicate these intervals visually, each segment of the function is drawn only within its defined domain section. Open circles are placed at the endpoints \(-2\) and \(2\) to show these values are not part of the function (since the function doesn't include \(-2\) and \(2\) themselves).

Plotting helps reveal how the function behaves visually, providing a clear depiction of its characteristics, such as where it's defined and how the output (range) looks across its domain. This visual approach solidifies understanding of the function's behavior across different input values.