Evaluate Piecewise Function At X = -1
Hey guys! Today, we're diving into the world of piecewise functions and tackling a specific problem. We've got a piecewise function, and our mission, should we choose to accept it, is to evaluate it at x = -1. Piecewise functions might seem a little intimidating at first, but trust me, they're totally manageable once you break them down. So, let's get started and see how we can solve this! We will explore the intricacies of piecewise functions, offering a step-by-step guide on how to evaluate them accurately. Whether you're a student grappling with this concept or simply looking to brush up on your math skills, this article will provide you with a clear and concise explanation. So, grab your calculators and let’s get started!
Understanding Piecewise Functions
Before we jump into evaluating the function at x = -1, 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 split into different pieces, each with its own set of rules. Each “piece” of the function is defined over a specific interval of the input (x) values. This means that depending on the value of x you're plugging in, you'll use a different formula to calculate the output (f(x)).
The key thing to remember with piecewise functions is that you need to identify which interval your input value falls into. That will tell you which piece of the function's definition you should use. It's like following a set of instructions, but the instructions change based on where you are. So, the first step in dealing with these functions is always to figure out which rule applies to your particular x-value. Understanding this concept is crucial for correctly evaluating piecewise functions, and it lays the groundwork for more advanced topics in calculus and analysis. Remember, each piece of the function behaves differently, so pay close attention to the intervals and their corresponding formulas.
The Given Piecewise Function
Okay, let's take a closer look at the specific piecewise function we're dealing with today. We've got this function, f(x), defined as follows:
What does this mean? Well, it's saying that if x is less than -4, we use the first rule: f(x) = -x² - x - 3. But, if x is greater than or equal to -1, we switch gears and use the second rule: f(x) = (1/3)x - 6. Notice how each piece has its own condition attached to it. This is super important because it tells us exactly when to use each formula. Piecewise functions are like having multiple functions packed into one, each with its own domain and behavior. To master these functions, you need to be comfortable interpreting these conditions and applying the correct rule for each input value. This careful attention to detail will ensure accurate evaluation and a solid understanding of how piecewise functions work.
Evaluating at x = -1
Now, for the main event: we need to evaluate this function at x = -1. So, the big question is: which piece of the function do we use? To figure this out, we need to see where x = -1 fits into the conditions we talked about earlier. Is -1 less than -4? Nope. Is -1 greater than or equal to -1? You bet! So, x = -1 falls into the second category, meaning we're going to use the second rule: f(x) = (1/3)x - 6. This is the crucial step in evaluating piecewise functions – identifying the correct piece to use. It's like choosing the right tool for the job; using the wrong formula will lead to the wrong answer. Once you've determined the appropriate piece, the rest is just plugging in the value and simplifying. So, always double-check the conditions before proceeding with the calculation to ensure accuracy and avoid common mistakes. This careful approach will make evaluating piecewise functions a breeze!
Plugging in the Value
Alright, we've established that we're using the rule f(x) = (1/3)x - 6 because x = -1 satisfies the condition x ≥ -1. Now, it's time to plug in the value of x into this formula. We simply replace x with -1, giving us:
f(-1) = (1/3)(-1) - 6
See? It's not so scary! We're just taking the value we want to evaluate the function at and substituting it into the appropriate expression. This is a fundamental skill in mathematics, and it's essential for working with all kinds of functions, not just piecewise ones. The key is to be meticulous and make sure you're replacing the variable (x in this case) with the correct value. Once you've done the substitution, the next step is just simplifying the expression, which is where our arithmetic skills come into play. So, let's move on to the simplification and see what our final answer will be!
Simplifying the Expression
Now, let's simplify the expression we got after plugging in x = -1:
f(-1) = (1/3)(-1) - 6
First, let's deal with the multiplication: (1/3)(-1) = -1/3. So, our expression becomes:
f(-1) = -1/3 - 6
To subtract these, we need a common denominator. We can rewrite 6 as a fraction with a denominator of 3: 6 = 18/3. Now we have:
f(-1) = -1/3 - 18/3
Finally, we can subtract the fractions: -1/3 - 18/3 = -19/3.
So, we've simplified the expression step by step, using basic arithmetic operations. This process of simplifying is crucial in mathematics, as it allows us to express our answers in the most concise and understandable form. Remember, breaking down the problem into smaller, manageable steps makes it easier to tackle and reduces the chances of making errors. In this case, we handled the multiplication first, then found a common denominator to subtract the fractions. This methodical approach is a valuable skill for any math problem, and it will help you confidently solve even the most complex equations.
The Final Answer
Alright, after all that plugging in and simplifying, we've arrived at our final answer! We found that:
f(-1) = -19/3
So, the value of the piecewise function f(x) at x = -1 is -19/3. And there you have it! We've successfully evaluated a piecewise function at a specific point. Remember, the key to tackling these types of problems is to carefully identify which piece of the function applies based on the given x-value, and then follow the rules for that piece. This step-by-step approach will help you navigate the sometimes-tricky world of piecewise functions with confidence. Congratulations on making it through this problem! Keep practicing, and you'll become a piecewise function pro in no time. Math can be fun and rewarding when you break it down and approach it methodically. So, keep exploring, keep learning, and keep solving!