Piecewise Function Analysis: Graphing And Limit Evaluation

Hey guys! Today, we're diving into the fascinating world of piecewise functions. We've got a juicy problem here that involves sketching a graph and figuring out some limits. Don't worry, we'll break it down step by step so it's super easy to follow. Let's get started!

Understanding Piecewise Functions

Before we jump into the specifics, let's make sure we're all on the same page about what a piecewise function actually is. Think of it like a function that's been broken up into different pieces, each with its own rule. The function behaves differently depending on the input value, x. It's like having a set of instructions, and the instruction you follow depends on where you are on the number line.

In our case, we have a function f(x) that's defined by three different expressions, each applying to a specific interval of x values:

1. For x less than 0, f(x) is equal to -x².
2. For x between 0 (inclusive) and 1 (exclusive), f(x) is equal to x² + 1.
3. For x greater than or equal to 1, f(x) is equal to -x + 3.

Each of these expressions represents a different "piece" of our function. To really grasp what's going on, we need to visualize it. That's where graphing comes in. But first, let's understand why piecewise functions are so important. Piecewise functions are extremely useful in modeling real-world situations where different rules or conditions apply over different intervals. Think about things like tax brackets (where the tax rate changes depending on your income), shipping costs (where prices might change based on weight or distance), or even the behavior of a thermostat (which turns on or off the heat based on temperature thresholds). Understanding these functions helps us to analyze and predict outcomes in many practical scenarios. So, stick with me, and let’s unravel this piecewise puzzle!

(a) Sketching the Graph of the Function

The first part of our mission is to sketch the graph of this piecewise function. Graphing a piecewise function might seem daunting at first, but it's really just a matter of plotting each piece separately within its specified interval. We'll tackle each piece one by one, and then put it all together to see the complete picture. Remember, each piece has its own domain, so we'll only draw that piece within its designated x values. Let's break it down:

Piece 1: f(x) = -x² for x < 0

This piece is a parabola that opens downwards. The negative sign in front of the x² term flips the standard parabola upside down. Now, since this piece is only defined for x values less than 0, we're only interested in the left side of the parabola. To sketch this, it’s helpful to think about a few key points. When x is 0, the function value is 0. However, since the domain is x < 0, we'll use an open circle at (0, 0) to indicate that this point isn't actually included in this piece of the function. As x becomes more negative (e.g., -1, -2, -3), f(x) becomes more negative as well (-1, -4, -9), creating the familiar downward curve of a reflected parabola. So, you'll be sketching the left half of a downward-facing parabola.

Piece 2: f(x) = x² + 1 for 0 ≤ x < 1

This piece is another parabola, but this time it opens upwards. The "+ 1" shifts the entire parabola up by one unit. This piece is defined for x values between 0 (inclusive) and 1 (exclusive). At x = 0, the function value is 1, and we'll use a closed circle at (0, 1) because x = 0 is included in this interval. At x = 1, the function value would be 2, but since the domain is x < 1, we'll use an open circle at (1, 2) to show that this point is not included. For x values between 0 and 1, the parabola will curve upwards, starting from (0, 1) and approaching (but not quite reaching) the point (1, 2).

Piece 3: f(x) = -x + 3 for x ≥ 1

This piece is a linear function with a negative slope. The equation f(x) = -x + 3 represents a straight line with a slope of -1 and a y-intercept of 3. This piece is defined for x values greater than or equal to 1. At x = 1, the function value is 2, and we'll use a closed circle at (1, 2) because x = 1 is included in this domain. As x increases, the function value decreases (because of the negative slope). For example, at x = 2, f(x) = 1, and at x = 3, f(x) = 0. This piece will appear as a line sloping downwards to the right.

Putting It All Together

Now for the exciting part: combining all three pieces to create the complete graph of f(x)! Imagine superimposing the three sketches we've made. The left side will have a downward-facing parabola, the middle section will have a portion of an upward-facing parabola shifted up by one unit, and the right side will have a line sloping downwards. Pay special attention to the endpoints of each piece and the open/closed circles, as they show where the function is defined and where it transitions from one piece to another. You should see a clear picture of how the function behaves differently over different intervals of x. The graph gives us a visual representation of the piecewise function, which is super helpful for understanding its behavior and for tackling the next part: evaluating limits.

(b) Investigating Limits

Now that we have a visual representation of our function, let's dive into the concept of limits. Remember, a limit asks what value a function approaches as x gets closer and closer to a particular value, without necessarily reaching it. For piecewise functions, we need to be extra careful when evaluating limits at the points where the function changes its definition (these are often called "breakpoints"). In our case, the breakpoints are x = 0 and x = 1, so we'll focus our attention there. We need to consider both the left-hand limit (approaching from values less than the point) and the right-hand limit (approaching from values greater than the point). If these one-sided limits exist and are equal, then the overall limit exists at that point. If they're different, the limit does not exist. Ready? Let's get to it!

Limit at x = 0

To determine the limit as x approaches 0, we need to consider both the left-hand limit (denoted as x → 0⁻) and the right-hand limit (denoted as x → 0⁺).

• Left-hand limit (x → 0⁻): As x approaches 0 from the left (i.e., x values less than 0), we use the first piece of the function: f(x) = -x². So, we evaluate the limit of -x² as x approaches 0 from the left. Substituting x = 0 into -x², we get 0. Therefore, the left-hand limit is 0.
• Right-hand limit (x → 0⁺): As x approaches 0 from the right (i.e., x values greater than 0), we use the second piece of the function: f(x) = x² + 1. So, we evaluate the limit of x² + 1 as x approaches 0 from the right. Substituting x = 0 into x² + 1, we get 1. Therefore, the right-hand limit is 1.

Since the left-hand limit (0) is not equal to the right-hand limit (1), the limit of f(x) as x approaches 0 does not exist. This makes sense visually, too – if you look at the graph around x = 0, you’ll see that the function is approaching different y-values from the left and right.

Limit at x = 1

Now, let’s repeat this process for x = 1.

• Left-hand limit (x → 1⁻): As x approaches 1 from the left (i.e., x values less than 1), we still use the second piece of the function: f(x) = x² + 1. So, we evaluate the limit of x² + 1 as x approaches 1 from the left. Substituting x = 1 into x² + 1, we get 2. Therefore, the left-hand limit is 2.
• Right-hand limit (x → 1⁺): As x approaches 1 from the right (i.e., x values greater than 1), we use the third piece of the function: f(x) = -x + 3. So, we evaluate the limit of -x + 3 as x approaches 1 from the right. Substituting x = 1 into -x + 3, we get 2. Therefore, the right-hand limit is 2.

In this case, the left-hand limit (2) is equal to the right-hand limit (2). Therefore, the limit of f(x) as x approaches 1 exists, and its value is 2. Again, this lines up with our visual understanding from the graph – the function is approaching the same y-value from both sides as x gets closer to 1.

Conclusion

And there you have it! We've successfully sketched the graph of our piecewise function and investigated the limits at the breakpoints. We found that the limit at x = 0 does not exist due to differing left-hand and right-hand limits, while the limit at x = 1 exists and is equal to 2. The key takeaway here is that when dealing with piecewise functions, it's crucial to consider each piece separately and pay close attention to the intervals and breakpoints. By understanding the behavior of each piece and evaluating the one-sided limits, we can confidently analyze these fascinating functions. Hope this helped, and keep exploring the awesome world of math!