Is This Piecewise Function Continuous? A Detailed Analysis

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Hey guys! Let's dive into a fascinating question about function continuity. We're going to analyze a piecewise function and determine whether it's continuous. This is a crucial concept in calculus, and understanding it will help you tackle more complex problems later on. So, let's get started!

Understanding Continuity

Before we jump into the specific function, let's quickly recap what it means for a function to be continuous. In simple terms, a function is continuous at a point if you can draw its graph without lifting your pen. More formally, a function f(x) is continuous at a point x = a if the following three conditions are met:

  1. f(a) is defined (the function exists at that point).
  2. The limit of f(x) as x approaches a exists.
  3. The limit of f(x) as x approaches a is equal to f(a).

If any of these conditions are not met, the function is discontinuous at that point. There are different types of discontinuities, such as removable discontinuities (holes), jump discontinuities, and infinite discontinuities (asymptotes). We'll be looking for these as we analyze our function.

The Piecewise Function

The function we're going to analyze is defined as follows:

f(x) = 
  \begin{cases}
    x^2 + 1 & \text{if } x < 1 \\
    2 & \text{if } x = 1 \\
    7 - 5x & \text{if } x > 1
  \end{cases}

This is a piecewise function, meaning it's defined by different expressions over different intervals of the domain. For x less than 1, the function is defined as x² + 1. At x = 1, the function is defined as 2. And for x greater than 1, the function is defined as 7 - 5x. The crucial point we need to investigate for continuity is x = 1, where the function's definition changes. We will analyze the continuity of this piecewise function step by step.

Checking for Continuity at x = 1

To determine if the function is continuous, especially at the critical point x = 1, we need to check the three conditions of continuity that we discussed earlier. Let's go through them one by one.

1. Is f(1) Defined?

First, we need to check if f(1) is defined. Looking at the piecewise definition, we see that when x = 1, the function is defined as f(x) = 2. Therefore, f(1) = 2, and the first condition is met. This means there isn't a hole or undefined spot directly at x = 1.

2. Does the Limit of f(x) as x Approaches 1 Exist?

Next, we need to determine if the limit of f(x) as x approaches 1 exists. For this, we need to check the left-hand limit and the right-hand limit separately. If these two limits are equal, then the limit exists.

Left-Hand Limit

The left-hand limit is the limit as x approaches 1 from the left (i.e., x values less than 1). In this case, we use the definition f(x) = x² + 1. So, we need to find:

lim (x→1-) (x² + 1)

We can evaluate this limit by direct substitution: (1)² + 1 = 1 + 1 = 2. Therefore, the left-hand limit is 2.

Right-Hand Limit

The right-hand limit is the limit as x approaches 1 from the right (i.e., x values greater than 1). In this case, we use the definition f(x) = 7 - 5x. So, we need to find:

lim (x→1+) (7 - 5x)

Again, we can evaluate this limit by direct substitution: 7 - 5(1) = 7 - 5 = 2. Therefore, the right-hand limit is also 2.

Comparing Limits

Since the left-hand limit (2) is equal to the right-hand limit (2), the limit of f(x) as x approaches 1 exists and is equal to 2. This means that as we approach x = 1 from either side, the function values converge to the same point.

3. Is the Limit Equal to f(1)?

Finally, we need to check if the limit of f(x) as x approaches 1 is equal to f(1). We found that the limit is 2, and we know that f(1) = 2. Since these two values are equal, the third condition for continuity is also met. This is a crucial step to confirm if there's a discontinuity in this piecewise function.

Conclusion: Is the Function Continuous?

We've checked all three conditions for continuity at x = 1:

  1. f(1) is defined (f(1) = 2).
  2. The limit of f(x) as x approaches 1 exists and is equal to 2.
  3. The limit of f(x) as x approaches 1 is equal to f(1).

Since all three conditions are met, the function f(x) is continuous at x = 1. But remember, we also need to consider the other intervals where the function is defined.

Checking Continuity Over the Entire Domain

We've established that the function is continuous at the point where the definition changes, x = 1. Now, we need to consider the other parts of the domain: x < 1 and x > 1. This involves understanding the properties of the functions that define f(x) in these intervals.

For x < 1: f(x) = x² + 1

When x < 1, the function is defined as f(x) = x² + 1. This is a polynomial function (specifically, a quadratic function), and polynomial functions are continuous everywhere. This is a fundamental property in calculus. Since x² + 1 is a polynomial, it's continuous for all x less than 1. There are no breaks, jumps, or asymptotes in this part of the function.

For x > 1: f(x) = 7 - 5x

When x > 1, the function is defined as f(x) = 7 - 5x. This is a linear function, which is also a type of polynomial function. As we mentioned earlier, polynomial functions are continuous everywhere. Therefore, 7 - 5x is continuous for all x greater than 1. Like the quadratic part, there are no discontinuities in this section of the function.

Overall Continuity

We've shown that f(x) is continuous at x = 1, for x < 1, and for x > 1. Since the function is continuous across its entire domain, we can confidently conclude that the function f(x) is continuous.

Key Takeaways

  • Continuity is essential: Understanding continuity is crucial in calculus and other areas of mathematics.
  • Piecewise functions need careful analysis: When dealing with piecewise functions, always check the points where the definition changes.
  • Three conditions for continuity: Remember the three conditions for continuity at a point: f(a) must be defined, the limit as x approaches a must exist, and the limit must equal f(a).
  • Polynomials are your friends: Polynomial functions are continuous everywhere, which simplifies the analysis.

Final Answer

Yes, the function f(x) is continuous. We've rigorously checked the continuity at the crucial point x = 1 and also considered the continuity of the function over its entire domain. Analyzing function continuity often involves a step-by-step process, especially for piecewise functions.

I hope this detailed explanation helped you understand how to determine the continuity of a piecewise function. Keep practicing, and you'll become a pro at this in no time! Let me know if you have any more questions, guys!