Finding The Range Of Piecewise Function H(x) A Step By Step Guide

by Sharif Sakr 66 views

Hey guys! Today, we're diving into the fascinating world of piecewise functions and figuring out how to determine their range. We'll be looking at a specific example, so you can follow along and learn how to tackle these types of problems. Let's get started!

The Function h(x) Defined

Let's consider the function h(x), which is defined in two parts, making it a piecewise function. These functions are like chameleons, changing their behavior depending on the input value x. Here’s how our function looks:

h(x) = { x + 2, if x < 3
       { -x + 8, if x ≥ 3

This notation might seem a bit daunting at first, but it's quite straightforward once you break it down. The function h(x) has two distinct rules:

  • Rule 1: If x is less than 3, we use the rule h(x) = x + 2. This means we take the input x, add 2 to it, and that's our output.
  • Rule 2: If x is greater than or equal to 3, we switch to the rule h(x) = -x + 8. Here, we take the negative of x, add 8, and that gives us our output.

Understanding these rules is crucial for determining the range of the function, which is the set of all possible output values. Now, let’s delve deeper into each part of the function to see what values they can produce.

Analyzing the First Piece: x + 2 for x < 3

Okay, let’s zoom in on the first part of our function: h(x) = x + 2 when x < 3. This is a linear function, which means it will produce a straight line when graphed. But we're only interested in the part of the line where x is less than 3.

Think about it this way: as x gets closer and closer to 3 (but never actually reaches it), what happens to h(x)? If x is 2.9, h(x) is 4.9. If x is 2.99, h(x) is 4.99. See the pattern? As x approaches 3, h(x) approaches 3 + 2 = 5. However, since x is strictly less than 3, h(x) will never actually equal 5. Instead, it will get infinitely close to 5 from below. We can say that the upper bound for this part of the function is 5, but it's an open bound, meaning 5 is not included in the range for this piece.

On the other hand, as x becomes a very large negative number (think -100, -1000, etc.), h(x) will also become a large negative number. There's no lower limit to how small x can be, so h(x) can also be infinitely small. This means that the range for this piece of the function extends towards negative infinity. So, for the first piece, the range is all values less than 5, or (-∞, 5).

Analyzing the Second Piece: -x + 8 for x ≥ 3

Now, let’s shift our focus to the second piece of the function: h(x) = -x + 8 when x ≥ 3. This is another linear function, but notice the negative sign in front of the x. This means the line will be sloping downwards, not upwards.

When x is exactly 3, h(x) = -3 + 8 = 5. So, this piece of the function starts at the value 5. Now, what happens as x gets larger and larger? If x is 4, h(x) is 4. If x is 10, h(x) is -2. As x increases, h(x) decreases because of the negative sign. There's no upper limit to how large x can be, so h(x) can become a very large negative number. This means that the range for this piece of the function extends towards negative infinity as well.

So, for the second piece, the function starts at 5 (when x = 3) and goes all the way down to negative infinity. The range for this piece is (-∞, 5]. Notice that 5 is included in the range here because the inequality is greater than or equal to 3.

Determining the Overall Range of h(x)

We've now analyzed both pieces of the function separately. To find the overall range of h(x), we need to combine the ranges of the individual pieces. Remember, the range is the set of all possible output values.

  • The first piece (x + 2 for x < 3) has a range of (-∞, 5).
  • The second piece (-x + 8 for x ≥ 3) has a range of (-∞, 5].

Notice something important: both pieces include all values less than 5. The first piece gets infinitely close to 5 but doesn't include it, while the second piece does include 5. This means that when we combine the ranges, we get all real numbers less than or equal to 5. Therefore, the overall range of h(x) is (-∞, 5].

Visualizing the Range

Sometimes, it helps to visualize what we've just discussed. Imagine graphing the function h(x). You'd see two lines connected at the point where x = 3. The first line (x + 2) would extend upwards towards (but not including) y = 5, and downwards towards negative infinity. The second line (-x + 8) would start at y = 5 and slope downwards towards negative infinity. The highest point on the graph is y = 5, and the graph covers all y-values below that. This visual representation reinforces our conclusion that the range is (-∞, 5].

Conclusion

So, guys, we've successfully navigated the world of piecewise functions and determined the range of our example function, h(x). Remember, the key is to break down the function into its individual pieces, analyze each piece separately, and then combine the results. Understanding how each piece behaves helps you understand the overall behavior of the function and its range.

I hope this explanation was helpful! Keep practicing with different piecewise functions, and you'll become a pro at finding their ranges in no time. Happy problem-solving!