Evaluating A Piecewise Function: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of piecewise functions. Don't let the name scare you; they're actually pretty straightforward once you get the hang of them. A piecewise function is just a function defined by multiple sub-functions, each applying to a certain interval of the main function's domain. We're going to break down how to evaluate a specific piecewise function, step by step, so you can tackle these problems with confidence.

Understanding Piecewise Functions

Before we jump into the calculations, let's make sure we understand what a piecewise function is all about. Imagine you have a function that behaves differently depending on the input value. It's like a choose-your-own-adventure book, but with math! You look at the input, see which condition it satisfies, and then use the corresponding formula to get the output. In our case, we have the following piecewise function:

$f(x)=\left\{\begin{array}{ll}x^2 & \text { if } x<4 \\ 5 x-6 & \text { if } x \geq 4\end{array}\right.$

This function, f(x), has two rules:

• If x is less than 4, we use the rule f(x) = x².
• If x is greater than or equal to 4, we use the rule f(x) = 5x - 6.

Key takeaway: The value of x determines which piece of the function we use.

To really nail this down, remember that the domain is split into intervals, and each interval has its own defining function. The conditions, like x < 4 or x ≥ 4, are super important because they tell you which "piece" of the function to use for a given x value. Getting this right is the most important part of evaluating piecewise functions, so take your time and double-check!

Now, with a firm grasp of what a piecewise function is, we can tackle the specific evaluations.

(a) Evaluating f(8)

Alright, let's find f(8). The key here is to figure out which condition x = 8 satisfies. Is 8 less than 4? Nope. Is 8 greater than or equal to 4? Yes! So, we use the second rule, which is f(x) = 5x - 6.

Plug in x = 8 into the second equation:

f(8) = 5(8) - 6

Now, let's simplify:

f(8) = 40 - 6

f(8) = 34

So, f(8) = 34. Easy peasy!

In summary:

1. We identified that x = 8 satisfies the condition x ≥ 4.
2. We used the corresponding function f(x) = 5x - 6.
3. We plugged in x = 8 and simplified to get f(8) = 34.

Now, let's move on to the next one. With practice, you will be able to quickly identify where each value of x falls, and apply the correct formula. Remembering the order of operations is crucial. Also, pay close attention to the inequality signs; a small mistake can lead to choosing the wrong function and getting the wrong answer. Keep an eye on the details, and you'll ace every piecewise function problem that comes your way.

(b) Evaluating f(-9)

Next up, let's evaluate f(-9). First, we need to determine which condition x = -9 satisfies. Is -9 less than 4? Yes, it is! Therefore, we'll use the first rule, which is f(x) = x².

Plug in x = -9 into the first equation:

f(-9) = (-9)²

Now, let's simplify:

f(-9) = 81

So, f(-9) = 81. Great!

To recap:

1. We identified that x = -9 satisfies the condition x < 4.
2. We used the corresponding function f(x) = x².
3. We plugged in x = -9 and simplified to get f(-9) = 81.

Remember that when you square a negative number, the result is always positive. This is a common area where students make mistakes, so it's essential to double-check your work. Also, keep in mind that the function f(x) = x² produces different results for positive and negative values of x, so it's crucial to pay attention to the sign of x. Understanding these nuances will help you avoid errors and solve piecewise functions accurately.

(c) Evaluating f(8.4)

Now, let's find f(8.4). Which condition does x = 8.4 satisfy? Is 8.4 less than 4? No. Is 8.4 greater than or equal to 4? Yes! So, we use the second rule again, which is f(x) = 5x - 6.

Plug in x = 8.4 into the second equation:

f(8.4) = 5(8.4) - 6

Let's simplify this:

f(8.4) = 42 - 6

f(8.4) = 36

Therefore, f(8.4) = 36.

Quick summary:

1. We identified that x = 8.4 satisfies the condition x ≥ 4.
2. We used the corresponding function f(x) = 5x - 6.
3. We plugged in x = 8.4 and simplified to get f(8.4) = 36.

When you're working with decimals, it's a good idea to take your time and double-check your calculations. A small mistake with decimal placement can lead to a wrong answer. You can also use a calculator to help you with the arithmetic, but make sure you still understand the steps involved. Practice with more examples involving decimals, and you'll become more comfortable and confident in evaluating piecewise functions with decimal inputs.

(d) Evaluating f(-2.1)

Lastly, let's evaluate f(-2.1). Does x = -2.1 satisfy the condition x < 4? Yes, it does! Therefore, we use the first rule: f(x) = x².

Plug in x = -2.1 into the first equation:

f(-2.1) = (-2.1)²

Now, let's simplify:

f(-2.1) = 4.41

So, f(-2.1) = 4.41.

Let's summarize:

1. We identified that x = -2.1 satisfies the condition x < 4.
2. We used the corresponding function f(x) = x².
3. We plugged in x = -2.1 and simplified to get f(-2.1) = 4.41.

Remember, squaring a negative number gives you a positive result. So, even though x = -2.1 is negative, f(-2.1) is positive. Also, be careful when squaring decimals; it's easy to make a mistake with decimal placement. Double-check your calculations, and you'll be fine. Keep practicing with different piecewise functions, and you'll become a pro at evaluating them, no matter the input!

Conclusion

And there you have it! We've successfully evaluated the piecewise function for four different values of x. Remember, the key to success with piecewise functions is to carefully determine which condition the given x value satisfies, and then use the corresponding rule to find f(x). Always double-check your work, especially when dealing with negative numbers or decimals.

Key takeaways for evaluating piecewise functions:

• Identify the correct interval: Determine which condition x satisfies.
• Apply the corresponding function: Use the function associated with that interval.
• Substitute and simplify: Plug in the value of x and simplify the expression.
• Double-check: Ensure your calculations and choices are correct.

With practice, you'll become more comfortable and confident in evaluating piecewise functions. Keep up the great work, and happy problem-solving!