Finding F(-6) For F(x) = 2|x-1| - 6: A Step-by-Step Guide

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Hey guys! Today, we're diving into a math problem that might look a little intimidating at first, but trust me, it's totally manageable. We're going to figure out how to find the value of f(-6) when we're given the function f(x) = 2|x-1| - 6. Don't worry if you're not a math whiz – we'll break it down step by step so everyone can follow along. Let's jump right in!

Understanding the Function: f(x) = 2|x-1| - 6

Before we can find f(-6), it's crucial to understand what this function f(x) = 2|x-1| - 6 actually means. This is a function that involves an absolute value, which might seem tricky, but it's really not that bad. Let's break down each part:

  • f(x): This is just the name of our function. It tells us that we're dealing with a function that takes an input (which we call x) and gives us an output.
  • |x-1|: This is the absolute value part. Remember, the absolute value of a number is its distance from zero, so it's always non-negative. For example, |3| = 3 and |-3| = 3. Inside the absolute value, we have x - 1, which means we're taking our input x, subtracting 1 from it, and then finding the absolute value of the result.
  • 2|x-1|: This means we're taking the absolute value we just found and multiplying it by 2. This stretches the absolute value function vertically.
  • - 6: Finally, we're subtracting 6 from the whole thing. This shifts the entire function down by 6 units.

So, in plain English, this function says: "Take an input x, subtract 1, find the distance from zero, double it, and then subtract 6." Got it? Great! Now let's move on to finding f(-6).

Step-by-Step Guide to Finding f(-6)

Okay, so our mission is to find f(-6). This basically means we need to plug in -6 for x in our function and see what we get. Here’s how we do it step-by-step:

Step 1: Substitute -6 for x

This is the easiest part. We just replace every x in the function with -6. So, f(x) = 2|x-1| - 6 becomes f(-6) = 2|-6-1| - 6. See? Not too scary, right?

Step 2: Simplify Inside the Absolute Value

Next, we need to simplify what's inside the absolute value. We have -6 - 1, which is just -7. So, now we have f(-6) = 2|-7| - 6. We're getting closer!

Step 3: Calculate the Absolute Value

Remember, the absolute value of a number is its distance from zero. The distance of -7 from zero is 7, so |-7| = 7. Our equation now looks like this: f(-6) = 2(7) - 6. Looking good!

Step 4: Multiply

Now we need to do the multiplication: 2 times 7 is 14. So, we have f(-6) = 14 - 6. Almost there!

Step 5: Subtract

Finally, we subtract 6 from 14, which gives us 8. So, f(-6) = 8. And that's it! We've found our answer.

Therefore, f(-6) = 8

Wrapping Up the Calculation

Let's recap what we just did. We started with the function f(x) = 2|x-1| - 6 and wanted to find f(-6). We substituted -6 for x, simplified the expression step by step, and ended up with f(-6) = 8. You nailed it! This kind of problem is all about following the order of operations and taking it one step at a time. Don't let the absolute value scare you; it's just a way of making sure we're dealing with distances, which are always positive or zero.

Visualizing the Function

To really understand what's going on, it can be helpful to visualize the function f(x) = 2|x-1| - 6. This function is a V-shaped graph because of the absolute value. The vertex (the bottom point of the V) is at (1, -6). The graph is stretched vertically by a factor of 2, and it's shifted down by 6 units.

When we found f(-6) = 8, we were essentially finding the y-coordinate of the point on the graph where x = -6. If you were to plot the graph, you'd see that the point (-6, 8) lies on the graph of the function. Visualizing the function can give you a better intuition for what the equation is actually doing.

Common Mistakes to Avoid

When working with absolute value functions, there are a few common mistakes that people make. Here are some things to watch out for:

  • Forgetting the Order of Operations: Always remember to follow the order of operations (PEMDAS/BODMAS). Simplify inside the absolute value first, then multiply, and finally add or subtract.
  • Incorrectly Handling the Absolute Value: Remember that the absolute value makes everything inside it non-negative. Don't forget to take the absolute value before performing other operations.
  • Making Arithmetic Errors: Simple arithmetic mistakes can throw off your entire answer. Double-check your calculations, especially when dealing with negative numbers.

By being aware of these common mistakes, you can avoid them and increase your chances of getting the correct answer.

Why This Matters: Real-World Applications

Okay, so finding f(-6) might seem like just a math exercise, but these kinds of functions actually have real-world applications. Absolute value functions are used in situations where you're interested in the magnitude or distance of something, rather than its direction.

For example:

  • Distance: If you're calculating the distance a car travels from a certain point, you might use an absolute value function. Distance is always positive, so the absolute value ensures that you get a positive result, regardless of the direction the car is traveling.
  • Error: In engineering and science, absolute value is often used to calculate the error between a measured value and an expected value. The error is the absolute difference between the two values, so it's always positive.
  • Optimization: Absolute value functions can also be used in optimization problems, where you're trying to minimize or maximize some quantity. For example, you might use an absolute value function to minimize the difference between a predicted value and an actual value.

So, while this specific problem might seem abstract, the underlying concepts are used in a variety of practical applications. Pretty cool, huh?

Practice Makes Perfect

The best way to get comfortable with absolute value functions is to practice! Try working through some more examples, and don't be afraid to make mistakes. Mistakes are a natural part of the learning process. The more you practice, the more confident you'll become.

Here are a few practice problems you can try:

  1. Find f(3) for f(x) = 3|x+2| - 5.
  2. Find f(-1) for f(x) = |2x-1| + 4.
  3. Find f(0) for f(x) = -2|x-3| + 1.

Work through these problems step-by-step, and remember to double-check your answers. If you get stuck, go back and review the steps we discussed earlier. You've got this!

Conclusion: You've Got This!

So, there you have it! We've successfully found f(-6) for the function f(x) = 2|x-1| - 6. We broke down the function, worked through the steps, and even talked about some real-world applications. Remember, the key to solving these kinds of problems is to take it one step at a time and not be afraid of the absolute value. It's just a way of dealing with distances!

I hope this guide was helpful and made things a little clearer. Keep practicing, keep asking questions, and you'll be a math whiz in no time. You've got this, guys! And remember, math can be fun – especially when you break it down and conquer it step by step. Until next time, happy calculating!