Problem 61
Question
The domain of each piecewise function \(i s(-\infty, \infty)\) a. Graph each function. b. Use your graph to determine the function's range. $$f(x)=\left\\{\begin{array}{rll} 2 x & \text { if } & x \leq 0 \\ 2 & \text { if } & x>0 \end{array}\right.$$
Step-by-Step Solution
Verified Answer
The graph of the function consists of two parts: a line that passes through the origin and has a slope of 2 for all \(x \leq 0\), and a horizontal line at \(y = 2\) for all \(x > 0\). The range of the function is \(-\infty\) to \(\infty\), since there is no \(y\) value that the function cannot take on.
1Step 1: Graph the first part of the function
Since for \(x \leq 0\), \(f(x) = 2x\), plot this line on the graph for all \(x\) values less than or equal to 0.
2Step 2: Graph the second part of the function
Since for \(x > 0\), \(f(x) = 2\), plot this line on the graph for all \(x\) values greater than 0. This would be a horizontal line at \(y = 2\). Make sure this part of the function does not include the point where \(x = 0\) since it is not defined here.
3Step 3: Identify the range
The range is the set of all \(y\) values that the function takes on. From the graph, identify the minimum and maximum \(y\) values.
Key Concepts
Domain and Range of FunctionsPlotting Linear FunctionsPiecewise Function Analysis
Domain and Range of Functions
Understanding the domain and range of functions is a fundamental aspect of algebra that allows students to grasp the full picture of how functions behave. The domain of a function refers to the set of all possible input values (typically the 'x' values) for which the function is defined. In the case of a piecewise function such as the one presented, the domain is all real numbers, or in interval notation, \( (-\infty, \infty) \).
The range, on the other hand, is the set of all possible outputs (the 'y' values) the function can produce. Determining the range involves looking at the graph and identifying the lowest and highest 'y' values the function can reach. In a piecewise function where there are different expressions for different intervals of the domain, each segment can have a different impact on the range. For instance, in our function, the line \( f(x) = 2x \) would continue indefinitely in both positive and negative 'y' directions for \( x \leq 0 \), but for \( x > 0 \) the value of \( f(x) \) is constantly 2, which greatly affects the overall range.
In our exercise, once the graph of each segment has been drawn, the range would encompass all 'y' values from negative infinity up to, and including, the constant value of 2, since that is the highest value reached by the function.
The range, on the other hand, is the set of all possible outputs (the 'y' values) the function can produce. Determining the range involves looking at the graph and identifying the lowest and highest 'y' values the function can reach. In a piecewise function where there are different expressions for different intervals of the domain, each segment can have a different impact on the range. For instance, in our function, the line \( f(x) = 2x \) would continue indefinitely in both positive and negative 'y' directions for \( x \leq 0 \), but for \( x > 0 \) the value of \( f(x) \) is constantly 2, which greatly affects the overall range.
In our exercise, once the graph of each segment has been drawn, the range would encompass all 'y' values from negative infinity up to, and including, the constant value of 2, since that is the highest value reached by the function.
Plotting Linear Functions
Plotting linear functions, which is one of the most basic yet crucial skills in algebra, becomes slightly more complex when dealing with piecewise functions. A linear function is typically in the form \( y = mx + b \), where \( m \) represents the slope, and \( b \) represents the y-intercept. When plotting \( f(x) = 2x \) for \( x \leq 0 \), you plot a straight line that passes through the origin (0,0) and has a slope of 2, meaning for every one unit you move to the right (along the x-axis), you move two units up (along the y-axis).
Specifically, for a piecewise function, the approach involves plotting the linear components for the intervals they are defined. It's crucial to emphasize that each part of a piecewise function is plotted separately and may or may not connect seamlessly with other parts. For example, with a point where \( x=0 \), extra care should be taken to include or exclude that point as per the function's definition. In educational materials and homework help contexts, it's helpful to remind students to consider the entire function's domain when plotting each segment and to pay close attention to open and closed circles at the endpoints to reflect the domain accurately.
Specifically, for a piecewise function, the approach involves plotting the linear components for the intervals they are defined. It's crucial to emphasize that each part of a piecewise function is plotted separately and may or may not connect seamlessly with other parts. For example, with a point where \( x=0 \), extra care should be taken to include or exclude that point as per the function's definition. In educational materials and homework help contexts, it's helpful to remind students to consider the entire function's domain when plotting each segment and to pay close attention to open and closed circles at the endpoints to reflect the domain accurately.
Piecewise Function Analysis
Piecewise function analysis is a critical skill for students to master as it combines understanding piecewise defined functions, their domains, and ranges. When analyzing a piecewise function, students must evaluate each piece of the function independently over the specified interval. For example, the given function \( f(x) \) has two separate expressions: \( f(x) = 2x \), defined for \( x \leq 0 \) and \( f(x) = 2 \) for \( x > 0 \).
Students should acknowledge that at \( x = 0 \) the function switches from one rule to another. This often represents a 'break' or 'jump' in the graph, and it's essential to note this on the graph properly. It's beneficial to encourage students to look at the larger picture of how the different 'pieces' of the function come together to form the whole and to underscore that the function's value can change abruptly at the boundaries of the intervals.
Improved comprehension of piecewise functions involves examining both the algebraic form and the graphical representation to understand how the function behaves overall. By combining these two approaches, students can gain a more profound understanding of the concept and apply this knowledge more effectively in solving problems.
Students should acknowledge that at \( x = 0 \) the function switches from one rule to another. This often represents a 'break' or 'jump' in the graph, and it's essential to note this on the graph properly. It's beneficial to encourage students to look at the larger picture of how the different 'pieces' of the function come together to form the whole and to underscore that the function's value can change abruptly at the boundaries of the intervals.
Improved comprehension of piecewise functions involves examining both the algebraic form and the graphical representation to understand how the function behaves overall. By combining these two approaches, students can gain a more profound understanding of the concept and apply this knowledge more effectively in solving problems.
Other exercises in this chapter
Problem 61
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