Analysis Of Piecewise Function F(x) = {-5x+3 If X≤2, X-4 If X>2} Graph And Continuity
To effectively graph the piecewise function f(x) defined as:
We need to consider each piece separately and then combine them on the same coordinate plane. This involves understanding the domain and the function's behavior within each interval. Piecewise functions are composed of different function definitions over distinct intervals of the domain. To accurately represent such functions, we must analyze and graph each piece independently, ensuring we pay close attention to the transition points and the function's behavior around these points.
Let's start with the first piece of the function, which is defined as f(x) = -5x + 3 for x ≤ 2. This is a linear function with a slope of -5 and a y-intercept of 3. To graph this, we can identify two points within the interval x ≤ 2. First, let's evaluate the function at x = 2, which is the boundary point for this interval. This gives us f(2) = -5(2) + 3 = -10 + 3 = -7. So, the point (2, -7) is on the graph. Since x = 2 is included in this interval (due to the ≤ sign), we use a closed circle at this point to indicate that it is part of the graph. Next, we can choose another point within the interval, such as x = 0. Evaluating the function at x = 0 gives us f(0) = -5(0) + 3 = 3. Thus, the point (0, 3) is also on the graph. Now we can draw a line passing through these two points, but we only draw the part of the line where x ≤ 2. This is because this piece of the function is only defined for values of x less than or equal to 2. The graph will extend from the point (2, -7) towards the left, continuing infinitely in that direction, showing the function's behavior for all x values less than or equal to 2.
Now, let's consider the second piece of the function, which is defined as f(x) = x - 4 for x > 2. This is also a linear function, but with a slope of 1 and a y-intercept of -4. Again, we start by evaluating the function at the boundary point, x = 2. However, we need to be careful here because this piece of the function is defined only for x > 2, not x = 2. If we substitute x = 2 into the equation, we get f(2) = 2 - 4 = -2. So, the point (2, -2) is a key reference point, but since x = 2 is not included in this interval (due to the > sign), we use an open circle at this point to indicate that it is not part of the graph. This open circle signifies a discontinuity in the function at x = 2. To graph this piece, we need another point within the interval x > 2. Let's choose x = 4. Evaluating the function at x = 4 gives us f(4) = 4 - 4 = 0. So, the point (4, 0) is on the graph. We can now draw a line starting from the open circle at (2, -2) and passing through the point (4, 0), extending towards the right. This line represents the function's behavior for all x values greater than 2. By combining the graphs of both pieces, we obtain the complete graph of the piecewise function f(x). The graph consists of a line segment for x ≤ 2 and another line segment for x > 2, with a discontinuity at x = 2 represented by the open circle. The visual representation clearly shows how the function behaves differently in different intervals of its domain, highlighting the essence of piecewise functions.
Continuity is a fundamental concept in calculus and mathematical analysis, referring to the unbroken nature of a function's graph. A function is said to be continuous at a point if there are no breaks, jumps, or holes in its graph at that point. More formally, a function f(x) is continuous at a point x = a if three conditions are met: f(a) is defined (the function has a value at a), the limit of f(x) as x approaches a exists, and the limit of f(x) as x approaches a is equal to f(a). If any of these conditions are not satisfied, the function is said to be discontinuous at x = a. Understanding continuity is crucial because it allows us to make certain predictions about the function's behavior, such as whether it will have intermediate values between any two points.
To determine whether the given piecewise function is continuous, we need to analyze its behavior at the point where the function definition changes, which is x = 2. This is because, away from this point, each piece of the function is a linear function, and linear functions are continuous everywhere. The critical point for assessing continuity is where the two pieces meet, as this is where a discontinuity is most likely to occur. We need to check the three conditions for continuity at x = 2. First, we need to determine if f(2) is defined. According to the function definition, when x ≤ 2, f(x) = -5x + 3. So, f(2) = -5(2) + 3 = -10 + 3 = -7. Thus, f(2) is defined and equal to -7. Next, we need to check if the limit of f(x) as x approaches 2 exists. For this, we need to consider the left-hand limit and the right-hand limit separately. The left-hand limit is the limit of f(x) as x approaches 2 from values less than 2. In this case, we use the function definition f(x) = -5x + 3. So, the left-hand limit is lim (x→2-) (-5x + 3) = -5(2) + 3 = -7. The right-hand limit is the limit of f(x) as x approaches 2 from values greater than 2. Here, we use the function definition f(x) = x - 4. So, the right-hand limit is lim (x→2+) (x - 4) = 2 - 4 = -2. Since the left-hand limit (-7) is not equal to the right-hand limit (-2), the limit of f(x) as x approaches 2 does not exist. Therefore, the second condition for continuity is not satisfied. Because the limit does not exist at x = 2, the function f(x) is discontinuous at x = 2. This means there is a jump or break in the graph of the function at this point. Visually, this discontinuity is represented by the jump from the closed circle at (2, -7) on the first piece of the function to the open circle at (2, -2) on the second piece.
In conclusion, the function f(x) is not continuous at x = 2 due to the differing left-hand and right-hand limits. This discontinuity is a key characteristic of piecewise functions, highlighting the importance of analyzing their behavior at the points where the function definition changes. The function is continuous everywhere else because each piece is a linear function, which is inherently continuous. Understanding the concept of continuity and how to determine it for various types of functions is essential in calculus and beyond, as it has significant implications for various mathematical operations and applications.
In summary, the piecewise function
Consists of two linear pieces that are graphed separately over their respective domains. The key to graphing this function accurately lies in understanding how to represent each piece within its specified interval. For x ≤ 2, the function is f(x) = -5x + 3, which is a line with a negative slope, and we graph this segment up to and including the point where x = 2. For x > 2, the function is f(x) = x - 4, another line but with a positive slope, and we graph this segment starting just after x = 2, indicating the exclusion of the point at x = 2 with an open circle. This distinction is crucial in piecewise functions as it illustrates how the function's definition changes across different intervals of its domain. Analyzing the graph provides immediate insight into the function's behavior, such as its increasing or decreasing nature, and any discontinuities.
The continuity of the function is a critical aspect to examine, especially at the point where the function definition transitions from one piece to another. In this case, the potential point of discontinuity is at x = 2. The function is continuous at a point if the function is defined at that point, the limit exists at that point, and the limit is equal to the function's value at that point. For this piecewise function, we found that f(2) = -7 is defined, but the limit as x approaches 2 does not exist because the left-hand limit (-7) and the right-hand limit (-2) are different. This difference in limits means that there is a “jump” in the graph at x = 2, making the function discontinuous at this point. Understanding these limits and how they relate to the function's value is essential for determining continuity. The discontinuity at x = 2 is a key characteristic of this piecewise function and affects its overall behavior and properties.
Determining continuity is not just a mathematical exercise but also has practical implications in various fields, such as physics, engineering, and economics, where continuous functions are often used to model real-world phenomena. Discontinuities in functions can represent abrupt changes or breaks in a system, which may be critical to understand and predict. In the context of piecewise functions, discontinuities can arise at the points where the definition of the function changes, making these points crucial to analyze. By understanding the conditions for continuity and how to test them, we can gain a deeper understanding of the behavior of functions and their applicability in different contexts. This analysis of the piecewise function's graph and continuity not only reinforces fundamental concepts in calculus but also enhances the ability to apply these concepts to more complex problems and situations.
The piecewise function is not continuous because the left-hand limit and the right-hand limit at x = 2 are not equal.
