Graphing & Domain: Piecewise Function F(x)

Hey guys! Today, we're diving into the fascinating world of piecewise functions. We'll be dissecting a specific function, figuring out its graph, and pinpointing its domain. Get ready to explore how these functions behave differently across various intervals!

Understanding the Piecewise Function

Let's kick things off by examining the function we're working with. We have a function, let's call it f(x), that's defined in pieces, hence the name "piecewise." It looks like this:

• f(x) = 1 if x < 0
• f(x) = x + 1 if 0 ≤ x < 2
• f(x) = 3 if 2 ≤ x < 10

What does this actually mean? Well, it means that the output of the function, the y-value, depends on the x-value we plug in. For any x less than 0, the function always spits out 1. When x is between 0 (inclusive) and 2 (exclusive), the function acts like x + 1. And when x is between 2 (inclusive) and 10 (exclusive), the function outputs a constant value of 3. See? It's like three mini-functions stitched together!

This piecewise function is a great example of how functions don't always have to follow a single, smooth curve. They can be made up of different segments, each with its own unique behavior. This makes them incredibly versatile for modeling real-world situations where relationships change abruptly or in stages.

For instance, imagine a parking garage that charges different rates depending on how long you park. The first hour might cost one amount, the next few hours a different amount, and so on. A piecewise function could perfectly describe this pricing structure. Or think about tax brackets, where your tax rate changes as your income increases. Piecewise functions are ideal for representing such scenarios.

Now, before we jump into graphing this function, let's think a bit more about what each piece represents. The first piece, f(x) = 1 for x < 0, is a horizontal line. No matter how far to the left of 0 we go on the x-axis, the y-value will always be 1. The second piece, f(x) = x + 1 for 0 ≤ x < 2, is a linear function with a slope of 1 and a y-intercept of 1. This means it's a straight line that slants upwards as x increases. The third piece, f(x) = 3 for 2 ≤ x < 10, is another horizontal line, this time at y = 3. So, we have two flat sections and a slanted section – quite a diverse function!

Graphing the Piecewise Function

Alright, let's get visual! Graphing a piecewise function might seem a bit tricky at first, but trust me, it's totally manageable. The key is to graph each piece separately, paying close attention to the intervals where they're defined.

1. Piece 1: f(x) = 1 for x < 0

   This is a horizontal line at y = 1. But remember, it's only valid for x values less than 0. So, we draw a horizontal line extending to the left from the y-axis. Now, here's a crucial detail: at x = 0, we use an open circle because the function is not defined as 1 at x = 0. It's only defined up to but not including 0.

2. Piece 2: f(x) = x + 1 for 0 ≤ x < 2

   This is a straight line. To graph it, we can find two points. Let's use the endpoints of the interval. When x = 0, f(x) = 0 + 1 = 1. So, we have the point (0, 1). Since the function is defined at x = 0, we use a closed circle here. Next, let's consider x = 2. f(x) = 2 + 1 = 3. So, we have the point (2, 3). However, the function is not defined at x = 2 in this piece (it's defined in the next piece), so we use an open circle at (2, 3). Now, we just draw a line connecting the closed circle at (0, 1) to the open circle at (2, 3).

3. Piece 3: f(x) = 3 for 2 ≤ x < 10

   This is another horizontal line, this time at y = 3. It's valid for x values between 2 (inclusive) and 10 (exclusive). So, we draw a horizontal line starting at x = 2. Since the function is defined at x = 2 in this piece, we use a closed circle. The line extends to x = 10, but we use an open circle at x = 10 because the function is not defined there.

And there you have it! The graph of our piecewise function is a combination of these three segments: a horizontal line at y = 1 for x < 0, a slanted line from (0, 1) to (2, 3) for 0 ≤ x < 2, and another horizontal line at y = 3 for 2 ≤ x < 10. Notice the open and closed circles – they're crucial for indicating where the function is defined and where it isn't.

Graphing piecewise functions can feel like assembling a puzzle. Each piece has its own shape and boundaries, and you need to fit them together correctly to get the complete picture. But with a bit of practice, you'll become a piecewise graphing pro!

Identifying the Domain

Now that we've got the graph down, let's talk about the domain. Remember, the domain is the set of all possible x-values that the function can accept. In other words, it's all the x-values for which the function gives us a valid y-value.

Looking at our function, we see that it's defined for x < 0, 0 ≤ x < 2, and 2 ≤ x < 10. Notice how these intervals kind of link up? The first interval goes up to (but doesn't include) 0, the second interval starts at 0 and goes up to (but doesn't include) 2, and the third interval starts at 2 and goes up to (but doesn't include) 10.

So, if we combine these intervals, we see that the function is defined for all x-values less than 10. We can write this in interval notation as (-∞, 10). The parenthesis on the 10 indicates that 10 is not included in the domain.

Thinking about the domain is like asking,