Inverse Of F(x) = 512x^(9/7): Proving & Finding F⁻¹'(x)

by TextBrain Team 56 views

Hey guys! Today, we're diving deep into the fascinating world of functions, specifically focusing on how to prove a function has an inverse and, if it does, how to find the derivative of that inverse. We'll be tackling the function f(x) = 512x^(9/7) as our example. So, buckle up, and let's get started!

Proving a Function is One-to-One and Has an Inverse

First things first, to show that a function has an inverse, we need to prove that it's one-to-one (also known as injective). A function is one-to-one if it never takes on the same value twice. In simpler terms, if f(a) = f(b), then a must equal b. An incredibly handy way to determine if a function is one-to-one is by using the fact that strictly increasing or strictly decreasing functions are always one-to-one.

So, let's break this down for our specific function, f(x) = 512x^(9/7). Our main goal here is to demonstrate that this function is either always increasing or always decreasing across its domain. To do this, we'll need to find its derivative, f'(x), and analyze its sign. Remember, a positive derivative indicates an increasing function, while a negative derivative indicates a decreasing function. So let’s calculate the derivative. Using the power rule, we get:

f'(x) = 512 * (9/7) * x^((9/7)-1) = (4608/7) * x^(2/7)

Now, let's analyze this derivative. Notice that x^(2/7) is the seventh root of x squared. Squaring any real number results in a non-negative value, and the seventh root of a non-negative number is also non-negative. Therefore, x^(2/7) is always greater than or equal to zero. Furthermore, the constant factor (4608/7) is positive. Consequently, f'(x) = (4608/7) * x^(2/7) is greater than or equal to zero for all x in the domain of f(x). The domain of f(x) includes all real numbers because we can take any real number to the power of 9/7. Because the derivative is positive (except at x=0), we have proved that the function is increasing over its entire domain.

Since f'(x) is positive for all x except 0, where it equals 0, f(x) is an increasing function. And as we mentioned earlier, increasing functions are inherently one-to-one. Because f(x) is one-to-one, it has an inverse function, which we can denote as f⁻¹(x). This is a crucial step because it confirms that our quest to find the derivative of the inverse function is actually possible!

Finding the Formula for the Derivative of the Inverse Function

Now that we know f⁻¹(x) exists, let's figure out how to find its derivative, (f⁻¹)'(x). Here's where a nifty formula comes to our rescue. This formula directly relates the derivative of the inverse function to the derivative of the original function. It states:

(f⁻¹)'(x) = 1 / f'(f⁻¹(x))

This might look a bit intimidating at first, but it's quite manageable once we break it down. It essentially says that the derivative of the inverse function at a point x is the reciprocal of the derivative of the original function evaluated at f⁻¹(x). The most important part here is understanding that to use this formula, we also have to find the inverse function.

Before we can use this formula, we need to find the inverse function, f⁻¹(x). To do this, we'll follow these steps:

  1. Replace f(x) with y: y = 512x^(9/7)
  2. Swap x and y: x = 512y^(9/7)
  3. Solve for y: This is where we isolate y to get the inverse function. First, divide both sides by 512: x/512 = y^(9/7). Next, raise both sides to the power of 7/9 to get rid of the exponent on y: (x/512)^(7/9) = y. So we have found the inverse function.
  4. Replace y with f⁻¹(x): f⁻¹(x) = (x/512)^(7/9)

Now that we have f⁻¹(x), we can plug it into our formula for the derivative of the inverse. Remember that we already found f'(x) = (4608/7) * x^(2/7). So, we need to evaluate f'(f⁻¹(x)):

f'(f⁻¹(x)) = (4608/7) * [(x/512)(7/9)](2/7)

Let’s simplify this expression, focusing on simplifying the exponent of (x/512). The exponent of (x/512) is (7/9)*(2/7) = 2/9. Replacing this value in the above formula, we get:

f'(f⁻¹(x)) = (4608/7) * (x/512)^(2/9)

Now we can plug this back into our formula for (f⁻¹)'(x):

(f⁻¹)'(x) = 1 / [(4608/7) * (x/512)^(2/9)] = (7/4608) * [1 / (x/512)^(2/9)]

This expression is the derivative of the inverse function. To simplify it further, we can rewrite it as:

(f⁻¹)'(x) = (7/4608) * (512/x)^(2/9)

And there you have it! We've successfully found the formula for the derivative of the inverse function. This journey involved proving that the function is one-to-one, finding the inverse function itself, and then applying the formula to calculate its derivative.

Putting It All Together

Let's recap what we've accomplished. We started with the function f(x) = 512x^(9/7) and wanted to show that it has an inverse and find the derivative of that inverse. Here’s a quick rundown of the steps:

  1. Proved f(x) is one-to-one: We found f'(x) and showed it's positive (except at x=0), meaning f(x) is increasing and therefore one-to-one.
  2. Found the inverse function, f⁻¹(x): By swapping x and y and solving for y, we found f⁻¹(x) = (x/512)^(7/9).
  3. Calculated (f⁻¹)'(x): Using the formula (f⁻¹)'(x) = 1 / f'(f⁻¹(x)), we found (f⁻¹)'(x) = (7/4608) * (512/x)^(2/9).

Why This Matters

Understanding inverse functions and their derivatives is super important in calculus and beyond. This knowledge is useful in a wide range of applications, from solving equations to optimizing processes in physics, engineering, and economics. The relationship between a function and its inverse is a fundamental concept, and mastering it opens doors to more advanced topics. Plus, being able to find the derivative of an inverse function without explicitly differentiating the inverse itself (thanks to our handy formula) is a powerful tool in your mathematical arsenal.

Final Thoughts

So, there you have it! We've successfully navigated the world of inverse functions and their derivatives. Remember, the key is to break down the problem into smaller, manageable steps. By proving the function is one-to-one, finding the inverse, and applying the derivative formula, you can tackle even the most challenging inverse function problems. Keep practicing, and you'll become a pro in no time! This journey is all about mastering the fundamentals and building a strong foundation for more advanced mathematical concepts.