Evaluating Piecewise Functions Finding F(-5), F(-2), And F(7)

Hey everyone! Today, we're diving into the fascinating world of piecewise functions. These functions are like chameleons, changing their behavior depending on the input value. We've got a great example here, and we're going to break it down step-by-step. So, grab your thinking caps, and let's get started!

The Piecewise Function: A Quick Overview

Before we jump into solving, let's quickly understand what a piecewise function actually is. Imagine a function that's defined by different formulas over different intervals of its domain. That's essentially what we're dealing with here. Our function, f(x), has two distinct rules:

• When x is less than -2, we use the formula: f(x) = x² - 1
• When x is greater than or equal to -2, we use the formula: f(x) = 5x + 3

The key here is to figure out which rule applies based on the input value we're given. Think of it like a fork in the road – the value of x tells us which path to take.

a) Finding f(-5): Navigating the First Path

Alright, let's tackle the first part: finding f(-5). The most important step here is figuring out which piece of our function applies when x is -5. Remember, our function has two rules:

• f(x) = x² - 1 if x < -2
• f(x) = 5x + 3 if x ≥ -2

Since -5 is definitely less than -2, we know we need to use the first rule: f(x) = x² - 1. This is where the magic happens. We're going to substitute -5 for x in this equation:

f(-5) = (-5)² - 1

Now, let's simplify. Remember that squaring a negative number makes it positive. So, (-5)² is 25. That gives us:

f(-5) = 25 - 1

Finally, a little subtraction, and we arrive at our answer:

f(-5) = 24

So, when x is -5, the value of our function, f(x), is 24. See? It's like following a map – we chose the correct path (the correct rule) based on the value of x, and we arrived at our destination (the function's value).

b) Calculating f(-2): The Boundary Case

Next up, we need to find f(-2). This is an interesting case because -2 is the boundary between our two rules. It's the point where the function switches from one formula to the other. So, which rule do we use? Let's look back at our function definition:

• f(x) = x² - 1 if x < -2
• f(x) = 5x + 3 if x ≥ -2

Notice the "greater than or equal to" symbol (≥) in the second rule. This means that when x is exactly -2, we use the second rule: f(x) = 5x + 3. It's crucial to pay attention to these inequalities because they tell us which rule is in charge at the boundary points.

Now, let's substitute -2 for x in the second equation:

f(-2) = 5(-2) + 3

Multiply 5 by -2, and we get -10:

f(-2) = -10 + 3

Finally, add -10 and 3, and we find:

f(-2) = -7

So, at the boundary point x = -2, the function's value is -7. This highlights the importance of carefully reading the function's definition, especially around the points where the rules change.

c) Determining f(7): Cruising Along the Second Path

Last but not least, we need to find f(7). For this one, we're cruising along the second path of our piecewise function. Why? Because 7 is definitely greater than -2. Let's remind ourselves of our two rules one more time:

• f(x) = x² - 1 if x < -2
• f(x) = 5x + 3 if x ≥ -2

Since 7 is greater than -2, we use the rule f(x) = 5x + 3. Let's substitute 7 for x:

f(7) = 5(7) + 3

Multiply 5 by 7, and we get 35:

f(7) = 35 + 3

Finally, add 35 and 3, and we have our answer:

f(7) = 38

Therefore, when x is 7, the value of our function is 38. This further illustrates how piecewise functions behave – they follow different rules depending on the input, making them versatile tools in mathematics.

Piecewise Functions: Key Takeaways

Okay, guys, we've successfully navigated this piecewise function! Let's recap the key things we learned:

1. Piecewise functions are defined by different rules over different intervals.
2. The inequalities in the function definition tell us which rule to use for a given input value.
3. Boundary points (where the rules change) require extra care – always check the "greater than or equal to" or "less than or equal to" signs.
4. To evaluate a piecewise function, substitute the input value into the correct rule and simplify.

Understanding piecewise functions opens the door to more advanced mathematical concepts. They're used in various fields, from computer graphics to economics, to model situations where behavior changes abruptly.

Practice Makes Perfect: Test Your Knowledge

Now that we've worked through this example together, why not try some practice problems on your own? You can find plenty of resources online or in your math textbook. The more you practice, the more comfortable you'll become with these fascinating functions.

Remember, math isn't just about formulas and equations; it's about understanding the underlying concepts. And piecewise functions, once you get the hang of them, are actually pretty cool!

If you have any questions or want to explore more examples, feel free to leave a comment below. Keep exploring, keep learning, and most importantly, keep having fun with math!