Evaluating F(x) = X^3 - 6x + 1 At X = 6

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Hey guys! Let's dive into a fun math problem today. We're going to evaluate a function, which basically means we're going to plug in a specific value for x and see what the function spits out. In this case, our function is f(x) = x^3 - 6x + 1, and we want to find out what happens when x = 6. Sounds interesting, right? Let’s get started and break it down step-by-step!

Understanding the Function

Before we jump into the calculations, let's make sure we understand what the function f(x) = x^3 - 6x + 1 is all about. Functions are like little machines: you feed them a number (x), and they do some math to it and give you back another number (f(x)). Our function here has a few parts:

  • x^3: This means we take our input x and multiply it by itself three times (x * x * x).
  • -6x: This means we multiply our input x by -6.
  • +1: This means we add 1 to the result of the previous calculations.

So, to evaluate this function at x = 6, we're going to replace every x in the equation with 6 and then do the math. It's like following a recipe – just substitute the ingredients!

Step-by-Step Calculation

Okay, let's get our hands dirty with the actual calculation. We're going to plug in x = 6 into our function f(x) = x^3 - 6x + 1. Here's how it looks:

  1. Replace x with 6: f(6) = (6)^3 - 6(6) + 1

    See? We just swapped out the xs with 6s. Now we need to simplify this expression using the order of operations (PEMDAS/BODMAS – Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).

  2. Calculate the exponent: f(6) = 216 - 6(6) + 1

    Here, 6^3 (6 cubed) is 6 * 6 * 6, which equals 216.

  3. Perform the multiplication: f(6) = 216 - 36 + 1

    Next, we multiply -6 by 6, which gives us -36.

  4. Do the addition and subtraction (from left to right): f(6) = 180 + 1

    We subtract 36 from 216, resulting in 180.

  5. Final addition: f(6) = 181

    Finally, we add 1 to 180, giving us our final answer: 181.

So, when we plug in x = 6 into the function f(x) = x^3 - 6x + 1, we get f(6) = 181. Awesome!

Visualizing the Function

Sometimes, it helps to visualize what's going on with a function. Our function f(x) = x^3 - 6x + 1 is a cubic function, which means it has a curved shape when we graph it. The point we just calculated, (6, 181), is a single point on that curve. Imagine drawing this curve on a graph – at the x-coordinate of 6, the y-coordinate (or the value of the function) is way up at 181. This visualization can help you understand how the function behaves for different values of x.

Why This Matters

You might be wondering, “Okay, we calculated a number, but why is this important?” Well, evaluating functions is a fundamental skill in mathematics and has tons of applications in the real world. Functions are used to model all sorts of things, from the trajectory of a ball thrown in the air to the growth of a population. By evaluating a function at different points, we can understand how the thing we're modeling changes over time or under different conditions. For example:

  • Physics: Functions can describe the position of an object as a function of time. Evaluating the function at a specific time tells you where the object is at that moment.
  • Economics: Functions can model the cost of producing a certain number of items. Evaluating the function tells you the cost for a specific production level.
  • Computer Graphics: Functions are used to create curves and surfaces in 3D models. Evaluating the function helps determine the shape of the object.

Common Mistakes to Avoid

When evaluating functions, there are a few common mistakes people make. Let’s make sure we avoid them:

  1. Order of Operations: Always remember PEMDAS/BODMAS! Do exponents before multiplication, and multiplication before addition and subtraction. Messing up the order can lead to a completely wrong answer.
  2. Sign Errors: Be super careful with negative signs. When you're plugging in a negative number, make sure you handle the signs correctly in every step. For instance, if we had to evaluate at x = -6, the -6x term would become -6(-6), which is positive 36. Pay close attention to these details!
  3. Simple Arithmetic: It sounds silly, but it's easy to make a small arithmetic mistake, especially when dealing with larger numbers. Double-check your calculations to ensure you haven’t made any slips. It’s always a good idea to use a calculator to verify your work, especially in exams or important calculations.

Practice Makes Perfect

The best way to get comfortable with evaluating functions is to practice! Try evaluating our function f(x) = x^3 - 6x + 1 at different values of x, like x = 0, x = -2, or x = 3. You can also try evaluating different functions altogether. The more you practice, the easier it will become. You can even use online tools or graphing calculators to check your answers and visualize the functions.

Let's Summarize

Alright, guys, let's quickly recap what we've learned today:

  • We evaluated the function f(x) = x^3 - 6x + 1 at x = 6.
  • We plugged in 6 for x and followed the order of operations to get f(6) = 181.
  • We discussed why evaluating functions is important and how it's used in real-world applications.
  • We talked about common mistakes to avoid, like messing up the order of operations or making sign errors.
  • We emphasized the importance of practice to master this skill.

Evaluating functions is a crucial skill in math, and now you've got the tools to tackle it with confidence. Keep practicing, and you'll become a function-evaluating pro in no time! Remember, every step in math builds on the previous one, and understanding functions is a fantastic foundation for more advanced topics. So, keep up the great work, and don't hesitate to explore more complex functions and problems. Math is like a puzzle, and every piece you solve makes the bigger picture clearer!

If you have any questions or want to try another example, just let me know. Keep exploring and keep learning!