Finding G(x-6) For G(x) = 7/(2x^2-4): A Step-by-Step Guide

by Alex Johnson 59 views

Let's dive into this mathematical problem and figure out how to find g(x-6) when we know that g(x) is defined as 7 divided by (2x^2 - 4). It might seem a little tricky at first, but don't worry! We'll break it down into easy-to-follow steps. This guide will walk you through the process, ensuring you understand each stage and can confidently tackle similar problems in the future.

Understanding the Function g(x)

Before we jump into finding g(x-6), let's make sure we fully grasp what the function g(x) means. The function g(x) is defined as g(x) = 7 / (2x^2 - 4). This means that for any input value 'x', we first square it (x^2), then multiply the result by 2 (2x^2), subtract 4 (2x^2 - 4), and finally divide 7 by that whole expression. Think of it like a little machine: you feed it a number 'x', and it spits out a new number based on these instructions. So, understanding this basic function is crucial for solving the problem.

Breaking Down the Components

To truly understand g(x), let's look at each part individually:

  • x^2: This means 'x' multiplied by itself. For example, if x = 3, then x^2 = 3 * 3 = 9.
  • 2x^2: Here, we take the result of x^2 and multiply it by 2. Using our previous example, if x^2 = 9, then 2x^2 = 2 * 9 = 18.
  • 2x^2 - 4: Next, we subtract 4 from the result of 2x^2. So, if 2x^2 = 18, then 2x^2 - 4 = 18 - 4 = 14.
  • 7 / (2x^2 - 4): Finally, we divide 7 by the entire expression (2x^2 - 4). Using our example, if 2x^2 - 4 = 14, then g(x) = 7 / 14 = 0.5.

By breaking down the function into these smaller steps, we can see exactly what's happening and how the input 'x' affects the final output. This is a valuable technique for understanding any function in mathematics.

The Goal: Finding g(x-6)

Now that we understand g(x), our goal is to find g(x-6). This might look a bit more complex, but the key is to realize that we're simply replacing 'x' in the original function with the expression '(x-6)'. This is a fundamental concept in function manipulation, and mastering it opens doors to solving a wide range of mathematical problems. So, instead of feeding 'x' into our function machine, we're now feeding in '(x-6)'. This means every instance of 'x' in the original equation needs to be replaced with '(x-6)'.

What Does g(x-6) Really Mean?

Think of it this way: g(x) tells us what to do with the input 'x'. Similarly, g(x-6) tells us what to do with the input '(x-6)'. We're not changing the rules of the function; we're just changing the input. The function will still perform the same operations—squaring, multiplying by 2, subtracting 4, and dividing 7—but it will do so on the expression '(x-6)' instead of just 'x'. This subtle but crucial difference is the heart of solving this type of problem.

Step-by-Step Solution for g(x-6)

Let's walk through the steps to find g(x-6). Remember, our original function is g(x) = 7 / (2x^2 - 4). The key is to substitute '(x-6)' wherever we see 'x'.

Step 1: Substitution

Replace every 'x' in the function with '(x-6)'. This gives us:

g(x-6) = 7 / (2(x-6)^2 - 4)

This is the most important step! We've successfully replaced 'x' with '(x-6)', and now we have a new expression for g(x-6). The rest of the solution involves simplifying this expression.

Step 2: Expand (x-6)^2

We need to expand the term (x-6)^2. Remember that (x-6)^2 means (x-6) multiplied by itself: (x-6) * (x-6). We can use the FOIL method (First, Outer, Inner, Last) or the distributive property to expand this:

(x-6)(x-6) = x*x - 6x - 6x + 36 = x^2 - 12x + 36

So, (x-6)^2 simplifies to x^2 - 12x + 36. This is a common algebraic expansion, and it's helpful to be comfortable with it.

Step 3: Substitute the Expanded Form

Now, let's substitute the expanded form of (x-6)^2 back into our equation:

g(x-6) = 7 / (2(x^2 - 12x + 36) - 4)

We've replaced (x-6)^2 with its expanded form, making the expression a bit longer but also more manageable.

Step 4: Distribute the 2

Next, we need to distribute the 2 across the terms inside the parentheses:

2(x^2 - 12x + 36) = 2x^2 - 24x + 72

So, we multiply each term inside the parentheses by 2. This simplifies the expression further.

Step 5: Substitute the Distributed Form

Now, substitute this back into our equation:

g(x-6) = 7 / (2x^2 - 24x + 72 - 4)

We're getting closer to our final simplified form!

Step 6: Combine Like Terms

We have a constant term of 72 and another constant term of -4. Let's combine these:

72 - 4 = 68

Step 7: Final Simplified Form

Now we can write the final simplified form of g(x-6):

g(x-6) = 7 / (2x^2 - 24x + 68)

And that's it! We've successfully found g(x-6). This expression tells us exactly how the function g behaves when the input is (x-6).

Common Mistakes to Avoid

When working with function transformations like this, there are a few common pitfalls to watch out for:

  • Incorrect Substitution: The most common mistake is not substituting (x-6) correctly for every instance of 'x' in the original function. Make sure you replace 'x' consistently throughout the equation.
  • Expanding Incorrectly: Expanding (x-6)^2 incorrectly is another frequent error. Remember the FOIL method or the distributive property to ensure you get the correct expansion: x^2 - 12x + 36.
  • Order of Operations: Make sure you follow the order of operations (PEMDAS/BODMAS) when simplifying the expression. Distribute before adding or subtracting, and handle exponents before multiplication.
  • Forgetting Parentheses: Parentheses are crucial when substituting and expanding. For example, 2(x-6)^2 is very different from 2x-6^2. Always use parentheses to maintain the correct order of operations.

By being aware of these common mistakes, you can significantly improve your accuracy when solving these types of problems. Careful attention to detail and a systematic approach are key.

Practice Problems

To solidify your understanding, let's try a couple of practice problems. These will give you a chance to apply the steps we've learned and build your confidence.

  1. If f(x) = 3x^2 + 2x - 1, find f(x+2).
  2. If h(x) = 5 / (x^2 + 3), find h(x-1).

Work through these problems step-by-step, using the same method we outlined above. Remember to substitute, expand, distribute, and simplify. The more you practice, the more comfortable you'll become with these types of function transformations.

Conclusion

Finding g(x-6) when g(x) = 7 / (2x^2 - 4) might have seemed daunting at first, but by breaking it down into manageable steps, we've shown that it's a very solvable problem. The key is to understand the concept of function substitution and to be careful with your algebraic manipulations. By practicing these types of problems, you'll develop a strong foundation in function transformations, which is a crucial skill in mathematics. Remember to always double-check your work and be mindful of common mistakes. With a little practice and a lot of patience, you can master these concepts!

For further learning and resources on functions and algebraic manipulations, consider exploring Khan Academy's Algebra 2 course.