Problem 41 - Algebra A Combined Function

Question

Graph each piecewise-defined function. Use the graph to determine the domain and range of the function. $$ g(x)=\left\\{\begin{array}{ll} -2 & \text { if } \quad x \leq 0 \\ -4 & \text { if } \quad x \geq 1 \end{array}\right. $$

Step-by-Step Solution

Answer

Domain: \((- fty, 0] \cup [1, fty)\). Range: \{-4, -2\}.

Step 1: Understand the Function

The given function is a piecewise function defined for two different intervals. For \( x \leq 0 \), the function value is \(-2\). For \( x \geq 1 \), the function value is \(-4\). There are no values defined for \(0 < x < 1\).

Step 2: Plot the First Piece

For \( x \leq 0 \), the function is a constant \(-2\). This means that for all \( x \) less than or equal to 0, the function \( g(x)\) is simply a horizontal line at \( y = -2 \). We plot this line and put a solid dot at \((0, -2)\) since the point is included in this interval.

Step 3: Plot the Second Piece

For \( x \geq 1 \), the function is \(-4\). This indicates a horizontal line at \( y = -4 \) starting from \( x = 1 \) and extending to positive infinity. We plot a solid dot at \((1, -4)\) because this point is included in the interval.

Step 4: Analyze the Graph Continuity

The function graph is discontinuous between \(0\) and \(1\) as no values for \( g(x) \) are provided in this range. Therefore, there is a gap in the graph between these two points.

Step 5: Determine the Domain

The domain of the function is the set of all \( x \) for which the function is defined. From the graph, we see that the function is defined for \(x \leq 0\) and \(x \geq 1\), so the domain is \((-fty, 0] \cup [1, fty)\).

Step 6: Determine the Range

The range is the set of all possible \( g(x) \) values that the function can take. From the graph, \( y \) can be \(-2\) and \(-4\). So, the range is simply \{-4, -2\}.

Key Concepts

Domain of a Function

The **domain of a function** is a fundamental concept in mathematics. It refers to all the possible input values (commonly represented as \( x \)) for which a function is defined. Understanding the domain helps us know what kind of values we can put into the function without causing any errors.
For a piecewise function like \( g(x) \), each "piece" of the function may have its own domain. In this case, \( g(x) \) is defined for \( x \leq 0 \) in the first piece, and for \( x \geq 1 \) in the second piece.
This gives us the combined domain of the two separate pieces:

• For the piece where the function is \( -2 \), the domain is \( (-\infty, 0] \).
• For the piece where the function is \( -4 \), the domain is \( [1, \infty) \).

Thus, the overall domain of the piecewise function is the union of these two sets, written as \( (-\infty, 0] \cup [1, \infty) \). This tells us where each part of the function lives on the \( x \)-axis.

Range of a Function

The **range of a function** is the set of all possible outputs the function can produce. Essentially, it tells us the values that actually come out of the function after plugging in the values from its domain.
For the piecewise function given by \( g(x) \), the range can be identified directly from the function definition:

• When \( x \leq 0 \), \( g(x) = -2 \). So, \( -2 \) is a value in the range.
• For \( x \geq 1 \), \( g(x) = -4 \). Hence, \( -4 \) is also in the range.

Since these are the only outputs given the constraints in the definition, the range of \( g(x) \) is simply \( \{-4, -2\} \). Even though the domain includes many values, the range is limited to just these specific outputs due to the function's constant nature in each interval.

Graphing Functions

**Graphing functions** is a visual method to represent a function on the coordinate plane, making it easier to understand and analyze its behavior. A piecewise function like \( g(x) \) consists of distinct constant parts, each defined in its own interval.
To graph these kinds of functions:

• Begin by plotting each segment individually. For \( g(x) \), plot the horizontal line \( y = -2 \) for \( x \leq 0 \), placing a solid dot at the endpoint \((0, -2)\) to include it in the graph.
• Next, graph the second part, \( y = -4 \) for \( x \geq 1 \), beginning the line with a solid dot at \( (1, -4) \).
• Note any gaps. Here, there’s a lack of definition between \( 0 < x < 1 \), indicating a discontinuity visible by the gap between the two segments.

These graphical representations reveal both the range and overlap of the domain visually, reinforcing the understanding of piecewise-defined functions.

Discontinuous Functions

A **discontinuous function** is one where there are one or more "jumps" or breaks in the graph. This means that at certain points, the function can suddenly change its value, or it may not even be defined.
In the case of our piecewise function \( g(x) \), discontinuity occurs between \( 0 \) and \( 1 \) since there is no defined value for \( x \) in this interval:

• For \( x \leq 0 \), the function holds steady at \( -2 \).
• For \( x \geq 1 \), it jumps to \( -4 \).

Because there is no smooth transition between these parts, a gap forms in the graph. Such discontinuities are crucial to notice because they often affect mathematical operations like solving equations, limits, and integration. Understanding where a function is discontinuous tells us more about its nature and how to address it in various mathematical contexts.