Problem 49 - College Algebra Essentials

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 \(f(x)\) is \{3, -3\}

Step 1: Understand the Piecewise Function

The given piecewise function defines two cases. If \(x \leq -1\), \(f(x)=3\), and if \(x > -1\), \(f(x)=-3\). The point where the function changes its expression is \(x = -1\).

Step 2: Graph the Piecewise Function

To graph this function, draw a horizontal line at \(y = 3\) extending to \(-\infty\) until \(x = -1\), and include a filled circle at the point \((-1,3)\) because this is part of the function's domain (because \(x \leq -1\)). Then draw another horizontal line at \(y = -3\) starting just after \(x = -1\) and continuing to \(+\infty\), with an open circle at \((-1,-3)\) to indicate that \(x = -1\) is not included in this part of the domain (because \(x > -1\)).

Step 3: Determine the Function's Range

The range of a function consists of all possible output values (y-values). In this case, looking at the graph, it's evident that the function only produces two values: 3 for \(x \leq -1\) and -3 for \(x > -1\). Thus, the range of the function is \{3, -3\}.

Key Concepts

graphing piecewise functions

Piecewise functions can have different expressions depending on the input value (x). When graphing these functions, each piece of the function needs to be treated separately.
For example, consider a function that has different equations for different segments of the x-axis. The transition point between these segments is often referred to as the "break point". In our exercise, the break point is at \(x = -1\).
When plotting a piecewise function on a graph:

• Identify the break points. Here, \(x = -1\) is crucial because it dictates where the function's definition changes.
• For regions of \(x\) that include the break point, use a closed circle for inclusion (such as \(x \leq -1\) using the point \((-1,3)\)).
• For regions beyond the break point, use an open circle (such as \(x > -1\) using the point \((-1,-3)\)).

This allows us to clearly see where each piece applies.

domain and range

Understanding a piecewise function's domain and range is key to interpreting its behavior. The domain represents all the possible input values, while the range is the resulting output values the function can produce.
In our exercise, the domain is \((-\infty, \infty)\), meaning x can be any real number. This is typical for many basic piecewise functions since there isn't a restriction on \(x\) being any real value.
The range needs to be found by analyzing the different possible outputs of the function. From our example:

• For \(x \leq -1\), the function outputs \(3\).
• For \(x > -1\), it outputs \(-3\).

Thus, the range is \{3, -3\}, since these are the only y-values that the function can output.

step-by-step solutions

Breaking down the process into digestible stages is helpful for understanding piecewise functions.
Start by examining each piece separately to understand what the function does at different intervals of x. In our exercise, the function changes at \(x=-1\), which serves as a clear dividing line.
Let's walk through it step-by-step:

• Step 1: Identify the expressions used. For \(x \leq -1\), the function is constant at \(f(x) = 3\). For \(x > -1\), it switches to \(f(x) = -3\).
• Step 2: Graph each segment. Use a closed circle at \((-1,3)\) and an open circle at \((-1,-3)\) for the break point distinction.
• Step 3: Determine the range by observing the y-values from the graph: \{3, -3\}.

Secure understanding by practicing the graph with different functions and identifying break points and ranges.

function graph interpretation

Interpreting the graph of a piecewise function involves recognizing how different pieces of the function contribute to its entire scope.
This interpretation answers "what does the graph tell us about the behavior of the function?" Here, the function provides one constant output until a certain point, then shifts to another after that.
From the graph we made:

• The horizontal line at \(y=3\) tells us that for any \(x \leq -1\), \(f(x) = 3\).
• The horizontal line switches to \(y=-3\) for \(x > -1\), indicating a shift in behavior.

Furthermore, interpreting such a graph can help in understanding scenarios where different rules apply over different domains, which can be particularly useful in engineering, economics, or computer science contexts where conditions can affect outcomes.