Find The Inverse Function Of F(x) = (1/9)x + 2

by ADMIN 47 views

Hey guys! Today, we're diving into the fascinating world of inverse functions, specifically focusing on a linear function. We'll break down the process of finding the inverse, making it super easy to understand. So, let's get started!

The Challenge: Finding the Inverse of f(x) = (1/9)x + 2

Our mission, should we choose to accept it (and we do!), is to determine the inverse of the function f(x) = (1/9)x + 2. We're presented with four potential answers:

A. h(x) = 18x - 2 B. h(x) = 9x + 18 C. h(x) = 9x - 18 D. h(x) = 18x + 2

But before we jump into solving, let's take a moment to understand what an inverse function actually is. Think of a function like a machine: you feed it an input (x), and it spits out an output (f(x)). The inverse function is like a reverse machine – you feed it the output, and it gives you back the original input. Mathematically, if f(a) = b, then the inverse function, denoted as f⁻¹(x), should satisfy f⁻¹(b) = a. This fundamental property guides our steps in finding the inverse.

Unpacking the Essence of Inverse Functions

Before we solve the specific problem at hand, let's solidify our understanding of inverse functions. Imagine a function as a set of instructions. For example, f(x) = 2x + 1 takes an input, multiplies it by 2, and then adds 1. The inverse function needs to undo these operations in reverse order. So, instead of multiplying by 2 and adding 1, we would subtract 1 and then divide by 2. This intuitive understanding is key to grasping the concept of inverses.

More formally, the inverse function f⁻¹(x) of a function f(x) is a function that reverses the effect of f(x). This means that if we apply f(x) and then f⁻¹(x) (or vice versa), we end up back where we started. This can be expressed mathematically as:

  • f⁻¹(f(x)) = x for all x in the domain of f
  • f(f⁻¹(x)) = x for all x in the domain of f⁻¹

This property is the cornerstone of verifying whether a function is indeed the inverse of another. We'll use this later to confirm our solution. Now, let's delve into the step-by-step process of finding an inverse function.

The Step-by-Step Journey to Finding the Inverse

Okay, so how do we actually find the inverse of a function? Here's the breakdown, a roadmap to navigate the inverse function landscape:

  1. Replace f(x) with y: This is simply a notational change, making the equation easier to manipulate. We're essentially saying that the output of the function, which we call f(x), can also be represented by the variable y. This step sets the stage for the next crucial operation.

  2. Swap x and y: This is the heart of the inversion process! We're reflecting the function across the line y = x. Think of it like flipping the roles of input and output. What was the input (x) now becomes the output (y), and vice versa. This swap embodies the very definition of an inverse function – reversing the roles of input and output.

  3. Solve for y: Now we have a new equation, but y is likely tangled up with other terms. Our goal is to isolate y on one side of the equation. This usually involves algebraic manipulations like adding, subtracting, multiplying, and dividing. The key is to perform the same operations on both sides of the equation to maintain balance. This step unravels the relationship and expresses y in terms of x.

  4. Replace y with f⁻¹(x): This is the final flourish! We're replacing y with the notation for the inverse function, f⁻¹(x). This signifies that we've successfully found the function that reverses the effect of the original function. We've completed our journey!

With these steps in mind, we're ready to tackle the specific problem and find the inverse of f(x) = (1/9)x + 2. Let's put this plan into action!

Applying the Steps to Our Function: f(x) = (1/9)x + 2

Alright, let's put our knowledge to the test and find the inverse of f(x) = (1/9)x + 2. We'll follow our four-step roadmap, making sure to show each step clearly.

Step 1: Replace f(x) with y

This is a simple substitution, making our equation look a little different but meaning the exact same thing:

  • y = (1/9)x + 2

Step 2: Swap x and y

This is where the magic happens! We interchange the roles of x and y:

  • x = (1/9)y + 2

Notice how the x is now where the y used to be, and vice versa. This swap is crucial for finding the inverse function.

Step 3: Solve for y

Now comes the algebraic maneuvering. We need to isolate y on one side of the equation. First, let's subtract 2 from both sides:

  • x - 2 = (1/9)y

Next, to get rid of the fraction, we multiply both sides by 9:

  • 9(x - 2) = y

Distributing the 9 on the left side, we get:

  • 9x - 18 = y

So, we've successfully solved for y! It's now expressed in terms of x.

Step 4: Replace y with f⁻¹(x)

Finally, we replace y with the inverse function notation:

  • f⁻¹(x) = 9x - 18

And there we have it! We've found the inverse function of f(x) = (1/9)x + 2.

The Answer and Why It's Correct

Looking back at our options, we see that:

  • C. h(x) = 9x - 18

matches our result. So, the correct answer is C! But let's not just stop there. It's always a good idea to double-check our work, especially when dealing with inverse functions.

Verifying Our Solution: The Ultimate Test

To be absolutely sure we've found the correct inverse function, we can use the property we discussed earlier: f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. Let's test both of these to make sure our answer holds up.

Test 1: f⁻¹(f(x)) = x

First, we need to find f⁻¹(f(x)). This means we'll plug the original function, f(x) = (1/9)x + 2, into our calculated inverse function, f⁻¹(x) = 9x - 18:

  • f⁻¹(f(x)) = 9((1/9)x + 2) - 18

Now, let's simplify:

  • f⁻¹(f(x)) = 9(1/9)x + 9(2) - 18
  • f⁻¹(f(x)) = x + 18 - 18
  • f⁻¹(f(x)) = x

Great! The first test passes. Let's move on to the second test.

Test 2: f(f⁻¹(x)) = x

This time, we'll plug our inverse function, f⁻¹(x) = 9x - 18, into the original function, f(x) = (1/9)x + 2:

  • f(f⁻¹(x)) = (1/9)(9x - 18) + 2

Simplifying:

  • f(f⁻¹(x)) = (1/9)(9x) - (1/9)(18) + 2
  • f(f⁻¹(x)) = x - 2 + 2
  • f(f⁻¹(x)) = x

Excellent! The second test also passes. Since both tests are successful, we can confidently say that f⁻¹(x) = 9x - 18 is indeed the inverse function of f(x) = (1/9)x + 2.

Conclusion: Mastering Inverse Functions

So, guys, we've successfully navigated the world of inverse functions! We started with a specific problem – finding the inverse of f(x) = (1/9)x + 2 – and along the way, we:

  • Defined what an inverse function is and how it works.
  • Learned a step-by-step method for finding the inverse of a function.
  • Applied this method to our specific problem and found the answer.
  • Verified our solution using the fundamental property of inverse functions.

This journey highlights the importance of understanding the underlying concepts and having a systematic approach to problem-solving. Inverse functions might seem tricky at first, but with practice and a clear understanding of the steps, you can master them! Remember, the key is to swap x and y and then solve for y. And always, always verify your answer! Keep practicing, and you'll be an inverse function pro in no time!