Evaluating A Piecewise Function: Find F(-3), F(-1), F(3)

by ADMIN 57 views

Hey guys! 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're going to break down a specific piecewise function and figure out how to evaluate it at different points. So, let's get started and make piecewise functions a piece of cake!

Understanding Piecewise Functions

Before we jump into the problem, let's quickly recap what piecewise functions are all about. Think of them as functions defined by different formulas over different intervals of their domain. It's like having a set of rules, and the rule you use depends on where your input (x-value) falls. This makes them incredibly versatile for modeling real-world situations where behavior changes at specific thresholds. You'll often see them in scenarios like tax brackets (where the tax rate changes based on income) or step functions (like the cost of postage based on weight).

In essence, a piecewise function f(x) is defined by multiple sub-functions, each applicable over a specific interval. The key is to identify which interval your input value belongs to, and then use the corresponding sub-function to calculate the output. This might sound tricky, but it's actually quite straightforward once you get the hang of it. The notation might look a bit intimidating at first, with the curly braces and multiple expressions, but it's just a compact way of representing these conditional rules. We'll see how this works in practice as we tackle the problem at hand, so don't worry if it's not crystal clear just yet. The more examples you work through, the more comfortable you'll become with piecewise functions and their unique way of defining relationships.

The Piecewise Function

Here's the function we'll be working with:

f(x) = { 
  7/2 + 2x,   x ≤ -1
  -5 + (3x)/2,  -1 < x < 3
  1/4 * x,     x ≥ 3
}

This function f(x) is defined in three parts. Let's break down each part:

  1. For any x value less than or equal to -1, we use the formula 7/2 + 2x.
  2. If x is strictly greater than -1 but less than 3, we use the formula -5 + (3x)/2.
  3. Finally, for x values greater than or equal to 3, we use the formula 1/4 * x.

So, depending on the x value we plug in, we'll use a different formula to find the corresponding f(x) value. The domain of this piecewise function, or the set of all possible input values (x), is all real numbers, since every real number falls into one of the defined intervals. But the range, the set of all possible output values (f(x)), will depend on the behavior of each piece of the function. Graphing the function can often give you a good visual representation of both the domain and the range, and how the function behaves overall. Now that we understand how the function is structured, let's move on to the main task: evaluating the function at specific points.

Evaluating f(-3)

Our first task is to find the value of f(-3). To do this, we need to figure out which part of the piecewise function applies when x = -3. Remember, each part of the function is only valid for a specific range of x values. So, the crucial step is to identify the correct range for our input value.

Looking at our function definition, we see that the first part, 7/2 + 2x, is used when x ≤ -1. Since -3 is indeed less than or equal to -1, this is the formula we'll use. So, we've pinpointed the right piece of the puzzle! Now it's just a matter of plugging in the value and crunching the numbers.

Let's substitute x = -3 into the expression 7/2 + 2x:

f(-3) = 7/2 + 2(-3)

Now we just need to simplify this expression using basic arithmetic. First, we multiply 2 by -3, which gives us -6. Then we have:

f(-3) = 7/2 - 6

To subtract 6 from 7/2, we need to express 6 as a fraction with a denominator of 2. 6 is the same as 12/2, so we can rewrite the expression as:

f(-3) = 7/2 - 12/2

Now we have two fractions with the same denominator, so we can simply subtract the numerators:

f(-3) = (7 - 12) / 2

This simplifies to:

f(-3) = -5/2

So, we've successfully evaluated the function at x = -3. The key here was to correctly identify the relevant part of the piecewise function based on the input value. This is a crucial skill when working with these types of functions. Now, let's move on to the next point and repeat the process to find f(-1).

Evaluating f(-1)

Next up, we want to find f(-1). Again, the most important step is to determine which part of the piecewise function applies when x = -1. This is where paying close attention to the inequality signs is crucial.

Looking back at our function definition:

f(x) = { 
  7/2 + 2x,   x ≤ -1
  -5 + (3x)/2,  -1 < x < 3
  1/4 * x,     x ≥ 3
}

We see that the first part, 7/2 + 2x, is defined for x ≤ -1. This means that when x is equal to -1, we do use this part of the function. The second part, -5 + (3x)/2, is defined for x strictly greater than -1 (i.e., x > -1), so it doesn't apply in this case.

Now that we've identified the correct piece, let's substitute x = -1 into the expression 7/2 + 2x:

f(-1) = 7/2 + 2(-1)

Simplifying, we get:

f(-1) = 7/2 - 2

To subtract 2 from 7/2, we need to rewrite 2 as a fraction with a denominator of 2. 2 is the same as 4/2, so:

f(-1) = 7/2 - 4/2

Subtracting the numerators gives us:

f(-1) = (7 - 4) / 2

f(-1) = 3/2

So, f(-1) = 3/2. We've successfully navigated another piece of the piecewise function! Notice how the subtle difference in the inequality sign (≤ vs. <) made all the difference in choosing the correct sub-function. This highlights the importance of being precise when working with piecewise functions. Now, let's tackle the last value, f(3), and see which piece of the function we need to use this time.

Evaluating f(3)

Finally, let's find f(3). As with the previous evaluations, the key is to carefully determine which part of the piecewise function applies when x = 3. Let's take another look at the function definition:

f(x) = { 
  7/2 + 2x,   x ≤ -1
  -5 + (3x)/2,  -1 < x < 3
  1/4 * x,     x ≥ 3
}

We can see that the third part, (1/4)x, is defined for x ≥ 3. This means that when x is equal to 3, we use this part of the function. The second part, -5 + (3x)/2, is defined for x strictly less than 3 (i.e., x < 3), so it doesn't apply here. It's all about pinpointing the correct interval for our input value.

Now, let's substitute x = 3 into the expression (1/4)x:

f(3) = (1/4) * 3

This is a straightforward multiplication:

f(3) = 3/4

So, f(3) = 3/4. We've successfully evaluated the function at all three specified points! By carefully considering the intervals defined in the piecewise function, we were able to select the correct formula for each input value. This methodical approach is the key to confidently working with piecewise functions.

Final Results

Alright, we've done the work! Let's summarize our findings:

  • f(-3) = -5/2
  • f(-1) = 3/2
  • f(3) = 3/4

We successfully evaluated the piecewise function at x = -3, x = -1, and x = 3. Remember, the trick with piecewise functions is to always identify the correct interval for your x value before plugging it into the corresponding formula. If you keep that in mind, you'll be a piecewise function pro in no time!

Key Takeaways

Let's recap the main points we've learned today about evaluating piecewise functions. These are the key steps to remember whenever you encounter a similar problem:

  1. Understand the Definition: Make sure you thoroughly understand the definition of the piecewise function. Pay close attention to the intervals and the corresponding formulas for each interval. The inequalities used to define the intervals (≤, <, ≥, >) are crucial, as they determine which formula applies for a given input value.
  2. Identify the Correct Interval: This is the most important step! For each input value you're asked to evaluate, determine which interval it belongs to. This involves carefully comparing the input value to the interval boundaries defined in the function.
  3. Apply the Corresponding Formula: Once you've identified the correct interval, use the formula associated with that interval to calculate the function's value at the given input. This usually involves simple substitution and arithmetic.
  4. Double-Check Your Work: It's always a good idea to double-check your work, especially when dealing with multiple steps. Make sure you've correctly identified the interval, substituted the value into the correct formula, and performed the arithmetic accurately.

By following these steps, you can confidently tackle any piecewise function evaluation problem. Remember, practice makes perfect, so try working through more examples to solidify your understanding.

Piecewise functions might seem intimidating at first, but hopefully, this walkthrough has shown you that they're quite manageable once you understand the core concept. It's all about breaking down the function into its individual pieces and applying the right piece at the right time. Keep practicing, and you'll be a master of piecewise functions in no time!