Calculating Values Of The Function F(x) = 7|x-2|

Hey guys! Today, we're diving into the world of functions, specifically dealing with absolute values. We've got a cool function here: f(x) = 7|x - 2|. Our mission is to calculate the values of this function for different inputs. Let's break it down step-by-step and make it super easy to understand. We'll be finding f(0), f(2), f(-2), f(x+1), and f(x^2 + 2). So, buckle up, and let's get started!

Understanding the Function

Before we jump into calculations, let's make sure we understand what this function, f(x) = 7|x - 2|, is all about. The key part here is the absolute value, represented by the vertical bars | |. The absolute value of a number is its distance from zero, which means it's always non-negative. For example, |3| = 3 and |-3| = 3. So, whatever is inside those bars, we're only concerned with its magnitude (positive value). This is crucial to understanding how the function behaves. The 7 in front just means we're multiplying the absolute value by 7.

In simpler terms, f(x) takes an input x, subtracts 2 from it, then takes the absolute value of the result, and finally multiplies that by 7. This might seem a bit abstract, but when we start plugging in numbers, it'll become crystal clear. It's like a mathematical machine that takes an input, processes it, and spits out an output. The absolute value part ensures that the output is never negative, adding an interesting twist to the function's behavior. Remember, the absolute value essentially 'flips' any negative number inside the bars to its positive counterpart, while positive numbers stay positive.

Now, let's get our hands dirty with some actual calculations. We'll start with the simplest ones and then move on to the more complex ones. The goal is to get comfortable with how this function works, so we can tackle any input thrown our way. Think of it like learning a new language; once you understand the grammar (the function's rules), you can construct sentences (calculate outputs) easily. So, let's translate this mathematical language into concrete numbers!

Calculating f(0)

Alright, let's start with finding f(0). This means we're going to replace every x in our function with 0. Our function is f(x) = 7|x - 2|, so substituting x with 0 gives us f(0) = 7|0 - 2|. Now, let's simplify this step by step. First, we deal with what's inside the absolute value: 0 - 2 = -2. So, we now have f(0) = 7|-2|. Remember, the absolute value of a number is its distance from zero, so |-2| = 2. Therefore, f(0) = 7 * 2, and finally, f(0) = 14.

See? Not too shabby! We've successfully calculated the value of the function when x is 0. It's like we've just solved the first piece of a puzzle. We plugged in the input, followed the function's rules, and got our output. The key here is to take it one step at a time and not rush the process. It's all about carefully substituting, simplifying, and then arriving at the final answer. Each step is like a mini-puzzle in itself, and solving each one correctly leads us to the bigger solution.

This process highlights the beauty of functions: they provide a clear, predictable way to transform inputs into outputs. When we substitute 0 for x, we're essentially asking the function, "Hey, what happens when I give you 0?". And the function, following its own internal logic, tells us, "You get 14!". Understanding this input-output relationship is fundamental to understanding functions, and we've just nailed it for the first case. Let's keep this momentum going and tackle the next one!

Calculating f(2)

Next up, let's calculate f(2). Just like before, we're going to substitute x with 2 in our function f(x) = 7|x - 2|. So, we get f(2) = 7|2 - 2|. Let's break it down. First, we look at what's inside the absolute value: 2 - 2 = 0. This gives us f(2) = 7|0|. Now, the absolute value of 0 is simply 0, so we have f(2) = 7 * 0. And anything multiplied by 0 is 0, so f(2) = 0.

That was even simpler than the first one! We've found that when x is 2, the function f(x) outputs 0. This is an interesting result because it shows that there's a specific input that makes the entire function equal to zero. This is a key concept in functions – sometimes, there are special inputs that lead to special outputs. In this case, x = 2 is one such special input.

What's happening here is that when x = 2, the expression inside the absolute value becomes 0, and the absolute value of 0 is 0. This 0 then gets multiplied by 7, but 7 * 0 is still 0. It's like a chain reaction: a specific input triggers a sequence of operations that ultimately leads to a very specific output. This highlights how functions can be sensitive to their inputs, and even a slight change in the input can sometimes lead to a significant change in the output. Understanding these relationships is what makes working with functions so fascinating, and we've just uncovered another piece of the puzzle.

Calculating f(-2)

Now, let's move on to f(-2). We're going to do the same thing – substitute x with -2 in our function f(x) = 7|x - 2|. This gives us f(-2) = 7|-2 - 2|. First, let's simplify inside the absolute value: -2 - 2 = -4. So, we have f(-2) = 7|-4|. The absolute value of -4 is 4, so f(-2) = 7 * 4. Finally, 7 * 4 = 28, which means f(-2) = 28.

Awesome! We've calculated f(-2) and found it to be 28. This result further illustrates the impact of the absolute value in our function. Even though we plugged in a negative number (-2), the absolute value made the result inside the bars positive before we multiplied by 7. This is a classic example of how absolute value functions can "reflect" negative inputs into the positive realm.

Think about it: we started with -2, which is 2 units to the left of zero on the number line. But after the absolute value does its thing, we're effectively dealing with 4, which is 4 units to the right of zero. The function then amplifies this distance by a factor of 7, resulting in 28. This kind of transformation is what makes absolute value functions so interesting and useful in various mathematical contexts. They allow us to focus on the magnitude of a quantity without worrying about its sign, which can be incredibly helpful in many real-world applications. So, with f(-2) = 28 under our belts, let's continue our journey and explore what happens when we plug in more complex expressions.

Calculating f(x+1)

Okay, things are about to get a little more interesting! Let's calculate f(x + 1). This time, we're not plugging in a number, but an expression: x + 1. But don't worry, the process is the same. We're still substituting, but instead of replacing x with a number, we're replacing it with (x + 1). So, f(x) = 7|x - 2| becomes f(x + 1) = 7|(x + 1) - 2|. Now, let's simplify. Inside the absolute value, we have (x + 1) - 2. We can combine the constants: 1 - 2 = -1. So, we get f(x + 1) = 7|x - 1|.

And that's it! We've calculated f(x + 1). Notice that we didn't get a single number as an answer this time. Instead, we got another expression in terms of x. This is because we plugged in an expression in terms of x. The result f(x + 1) = 7|x - 1| tells us how the function behaves when its input is x + 1. It's like we've created a new function that's related to our original function. This is a powerful idea in mathematics: we can manipulate functions and create new ones based on the original.

What we've essentially done is shifted the input to our function. Instead of directly feeding x into the function, we're feeding x + 1. This shift has a predictable effect on the output, and by finding f(x + 1), we've quantified that effect. We now have a new expression that describes the function's behavior for this shifted input. This kind of manipulation is common in many areas of math and science, and understanding how to do it is a valuable skill. So, with f(x + 1) conquered, let's move on to the final challenge: calculating f(x^2 + 2).

Calculating f(x²+2)

Alright, our final challenge is here: calculating f(x^2 + 2). This looks a bit intimidating, but trust me, the process is the same as before. We're going to substitute x with the expression x^2 + 2 in our function f(x) = 7|x - 2|. So, we get f(x^2 + 2) = 7|(x^2 + 2) - 2|. Now, let's simplify. Inside the absolute value, we have (x^2 + 2) - 2. The 2 and -2 cancel each other out, leaving us with x^2. So, we have f(x^2 + 2) = 7|x^2|.

Now, here's a little trick: since x^2 is always non-negative (a square of any real number is non-negative), the absolute value doesn't actually change anything. In other words, |x^2| is the same as x^2. So, we can simplify further to f(x^2 + 2) = 7x^2.

And we're done! We've calculated f(x^2 + 2) and found it to be 7x^2. This is a neat result because it shows how our original function transforms when we feed it a quadratic expression. The absolute value, in this case, becomes almost irrelevant because x^2 is always positive or zero. This highlights an important point: sometimes, the specific form of the input can simplify the function's behavior.

By substituting x^2 + 2 for x, we've essentially created a new function that's a blend of our original function and the quadratic expression. The result, 7x^2, is a parabola, which is a very different shape from the original function, which involves an absolute value. This demonstrates how functions can be combined and transformed to create new mathematical objects with their own unique properties. With this final calculation, we've completed our mission. Let's take a moment to recap what we've learned.

Conclusion

So, guys, we've successfully navigated the function f(x) = 7|x - 2| and calculated its values for various inputs: f(0) = 14, f(2) = 0, f(-2) = 28, f(x + 1) = 7|x - 1|, and f(x^2 + 2) = 7x^2. We started by understanding the role of the absolute value, and then we systematically substituted each input, simplified the expression, and arrived at our results.

We saw how the absolute value affects the function's output, how special inputs can lead to special outputs (like f(2) = 0), and how plugging in expressions can lead to new expressions that describe the function's behavior for those inputs. We also learned a handy trick about how |x^2| is the same as x^2.

This exercise is a fantastic example of how functions work and how we can manipulate them. It's all about understanding the rules, applying them carefully, and breaking down complex problems into smaller, manageable steps. Functions are a fundamental concept in mathematics, and mastering them opens the door to a whole world of exciting ideas and applications. Keep practicing, keep exploring, and you'll become function-calculating masters in no time! You've got this! 🚀