Continuity Of Piecewise Function At X = -3: A Detailed Guide

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Determining the continuity of a function at a specific point is a fundamental concept in calculus. In this comprehensive guide, we will discuss how to determine whether the piecewise function f(x) is continuous at x = -3. The given function f(x) is defined differently for x greater than -3 and x less than or equal to -3. Understanding continuity is crucial for various applications in mathematics and other fields, so let’s dive in and break down the process step by step. We'll explore the conditions for continuity and apply them to the specific function provided, ensuring you grasp the core concepts and can tackle similar problems with confidence. This involves checking the function's limit from both sides and ensuring it matches the function's value at the point in question.

Understanding Continuity

Before we delve into the specifics of the given piecewise function, it’s essential to understand what continuity means in mathematical terms. A function f(x) is said to be continuous at a point x = c if it satisfies three key conditions:

  1. f(c) is defined: The function must be defined at the point c. In other words, when you plug c into the function, you should get a real number.
  2. The limit of f(x) as x approaches c exists: This means that as x gets closer and closer to c from both sides (left and right), the function f(x) approaches a specific value. Mathematically, this is written as lim β‚“β†’κœ€ f(x) exists.
  3. The limit of f(x) as x approaches c is equal to f(c): This condition ties the first two together. The value that the function approaches as x gets close to c must be the same as the function's value at c. Mathematically, this is expressed as lim β‚“β†’κœ€ f(x) = f(c).

If any of these three conditions are not met, the function is said to be discontinuous at x = c. Discontinuities can manifest in several ways, such as jumps, holes, or vertical asymptotes. Understanding these conditions is the first critical step in assessing the continuity of a function at a particular point. Each of these conditions plays a crucial role, ensuring that the function behaves predictably around the point of interest. The first condition ensures the function is defined, the second ensures the function approaches a single value from both directions, and the third ensures that the approached value aligns with the function's actual value at that point. Without all three conditions being met, the function cannot be considered continuous. Now that we have a solid grasp of the theoretical background, let's apply these concepts to our specific piecewise function and see how we can determine its continuity at x = -3.

The Piecewise Function

The piecewise function we are examining is defined as follows:

  • f(x) = 12 + xΒ², for x > -3
  • f(x) = 15 - 2x, for x ≀ -3

This function behaves differently depending on the value of x. When x is greater than -3, the function follows the rule 12 + xΒ², which represents a parabola opening upwards. When x is less than or equal to -3, the function follows the rule 15 - 2x, which represents a linear function. The critical point we are interested in is x = -3, where the function's definition changes. To determine continuity at this point, we need to check the three conditions mentioned earlier. The nature of piecewise functions makes them particularly interesting to study for continuity, as the transition points between the different definitions are where discontinuities are most likely to occur. In our case, x = -3 is where we need to scrutinize the function's behavior closely. This involves evaluating the function's value at x = -3, as well as examining the limits from both the left and the right sides. The behavior of the function on either side of x = -3 will dictate whether the function smoothly transitions at this point, which is the essence of continuity.

Step 1: Check if f(-3) is Defined

First, we need to determine if the function f(x) is defined at x = -3. According to the function's definition, when x is less than or equal to -3, f(x) = 15 - 2x. So, we use this part of the function to evaluate f(-3):

f(-3) = 15 - 2(-3) = 15 + 6 = 21

Thus, f(-3) is defined and equal to 21. This is the first step in our continuity check, and it passes the first condition. The fact that f(-3) has a specific value is a good start, but it is only the beginning. We still need to ensure that the function approaches this value as x gets closer to -3 from both sides. This involves examining the left-hand limit and the right-hand limit. If these limits exist and are equal to f(-3), then we can confidently say that the function is continuous at x = -3. If, however, either of these limits does not exist or is not equal to 21, then the function is discontinuous at this point. So, let's proceed to the next steps and carefully evaluate the limits.

Step 2: Check if the Limit Exists

To check if the limit of f(x) as x approaches -3 exists, we need to examine the left-hand limit and the right-hand limit separately. The left-hand limit (lim β‚“β†’-3⁻ f(x)) is the value that f(x) approaches as x approaches -3 from values less than -3. The right-hand limit (lim β‚“β†’-3⁺ f(x)) is the value that f(x) approaches as x approaches -3 from values greater than -3.

Left-Hand Limit

For the left-hand limit, we use the part of the function defined for x ≀ -3, which is f(x) = 15 - 2x:

lim β‚“β†’-3⁻ f(x) = lim β‚“β†’-3⁻ (15 - 2x) = 15 - 2(-3) = 15 + 6 = 21

So, the left-hand limit is 21. This tells us that as x gets closer to -3 from the left side, the function values approach 21. This is an important piece of the puzzle, but we need to see if the function behaves similarly as we approach from the right side. If the right-hand limit is different, it would indicate a jump discontinuity at x = -3. Therefore, let's move on to evaluating the right-hand limit.

Right-Hand Limit

For the right-hand limit, we use the part of the function defined for x > -3, which is f(x) = 12 + xΒ²:

lim β‚“β†’-3⁺ f(x) = lim β‚“β†’-3⁺ (12 + xΒ²) = 12 + (-3)Β² = 12 + 9 = 21

So, the right-hand limit is also 21. Since both the left-hand limit and the right-hand limit exist and are equal, we can conclude that the limit of f(x) as x approaches -3 exists and is equal to 21. This is a critical finding because it satisfies the second condition for continuity. The function approaches the same value from both directions, indicating a smooth transition at x = -3. However, we still have one more condition to verify. We need to ensure that this limit matches the function's actual value at x = -3, which we calculated in Step 1. This will confirm whether the function is truly continuous at this point or if there is a removable discontinuity. Let's proceed to the final step to complete our analysis.

Step 3: Check if the Limit Equals f(-3)

Finally, we need to check if the limit of f(x) as x approaches -3 is equal to f(-3). We have already found that:

  • lim β‚“β†’-3 f(x) = 21
  • f(-3) = 21

Since lim β‚“β†’-3 f(x) = f(-3) = 21, the third condition for continuity is satisfied. This is the final piece of the puzzle, and it confirms that our function is continuous at x = -3. The limit exists, the function is defined at the point, and the limit's value matches the function's value. This complete alignment is what defines continuity in mathematical terms. Therefore, we can now confidently conclude that the given piecewise function is indeed continuous at the specified point.

Conclusion

In conclusion, to determine if the piecewise function f(x) is continuous at x = -3, we followed a step-by-step process:

  1. We verified that f(-3) is defined and found it to be 21.
  2. We calculated the left-hand limit and the right-hand limit as x approaches -3, and both were equal to 21, confirming that the limit exists.
  3. We confirmed that the limit as x approaches -3 is equal to f(-3).

Since all three conditions for continuity are satisfied, we can definitively say that the function f(x) is continuous at x = -3. This systematic approach to checking continuity is applicable to a wide range of functions and is a fundamental skill in calculus. By understanding and applying these steps, you can confidently assess the continuity of any function at any given point. Understanding continuity is not just an academic exercise; it has practical implications in various fields, including physics, engineering, and computer science, where continuous functions often model real-world phenomena. For further exploration of continuity and related concepts, you can visit Khan Academy's Calculus section on Continuity.