Finding The Inverse Of F(x) = 2∛(3x - 1) - 5 A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of inverse functions. Specifically, we're going to tackle the function f(x) = 2∛(3x - 1) - 5 and walk through the process of finding its inverse, denoted as f⁻¹(x). This might seem a bit daunting at first, but trust me, with a clear, step-by-step approach, it's totally manageable. So, grab your thinking caps, and let's get started!

Understanding Inverse Functions

Before we jump into the nitty-gritty, let's take a moment to understand what an inverse function actually is. Simply put, an inverse function "undoes" what the original function does. Think of it like a mathematical U-turn. If f(x) takes an input x and transforms it into an output y, then f⁻¹(x) takes that output y and transforms it back into the original input x. This fundamental relationship is key to finding the inverse.

Mathematically, this relationship is expressed as follows:

• If f(a) = b, then f⁻¹(b) = a.

This means if we plug a value, say 'a', into our original function and get 'b' as the answer, then plugging 'b' into the inverse function should give us back 'a'. This 'undoing' action is the core concept behind inverse functions. Now, a crucial point to remember is that not every function has an inverse. For a function to have an inverse, it must be one-to-one. A one-to-one function means that each input value corresponds to a unique output value, and vice versa. Graphically, this can be checked using the horizontal line test: if any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one and does not have an inverse.

Luckily, the function we are dealing with, f(x) = 2∛(3x - 1) - 5, is a one-to-one function. Cube root functions, in general, are one-to-one, and the transformations applied here (multiplication by 2, horizontal stretch/compression, and vertical translation) don't change that property. So, we're good to go ahead and find its inverse!

Step 1: Replace f(x) with y

The first step in finding the inverse function is to replace f(x) with y. This is a simple notational change that makes the subsequent steps a bit clearer. So, we rewrite our function as:

y = 2∛(3x - 1) - 5

This might seem like a trivial step, but it helps us to think of the function in terms of input (x) and output (y), which is essential for the next step.

Step 2: Swap x and y

This is the heart of the inverse function finding process! We swap the positions of x and y in the equation. This reflects the idea that the inverse function reverses the roles of input and output. So, after swapping, our equation becomes:

x = 2∛(3y - 1) - 5

Notice that x is now expressed in terms of y, instead of the other way around. This is precisely what we need to do to find the inverse. Our goal now is to isolate y on one side of the equation. This will give us the inverse function in the familiar form of y = f⁻¹(x).

Step 3: Isolate y

Now comes the algebraic manipulation! Our mission is to get y all by itself on one side of the equation. We'll do this by carefully undoing the operations that are being applied to y, following the reverse order of operations (PEMDAS in reverse, often remembered as SADMEP). Currently, y is inside a cube root, being multiplied by 3, having 1 subtracted from it, the whole cube root is being multiplied by 2, and then 5 is being subtracted. So, let's undo these operations one by one.

3.1: Add 5 to both sides

The first thing we'll do is get rid of the -5. To do this, we add 5 to both sides of the equation:

x + 5 = 2∛(3y - 1)

3.2: Divide both sides by 2

Next, we want to get rid of the multiplication by 2. We do this by dividing both sides of the equation by 2:

(x + 5) / 2 = ∛(3y - 1)

3.3: Cube both sides

Now, we need to get rid of the cube root. The inverse operation of taking the cube root is cubing. So, we cube both sides of the equation:

((x + 5) / 2)³ = 3y - 1

3.4: Add 1 to both sides

We're getting closer! Now, we add 1 to both sides to isolate the term with y:

((x + 5) / 2)³ + 1 = 3y

3.5: Divide both sides by 3

Finally, we divide both sides by 3 to get y by itself:

[((x + 5) / 2)³ + 1] / 3 = y

Phew! That was a bit of algebraic maneuvering, but we did it! We've successfully isolated y.

Step 4: Replace y with f⁻¹(x)

The last step is simply a notational one. We replace y with f⁻¹(x) to indicate that we've found the inverse function. So, our final answer is:

f⁻¹(x) = [((x + 5) / 2)³ + 1] / 3

And there you have it! We've successfully found the inverse function of f(x) = 2∛(3x - 1) - 5.

Let's Summarize the Steps

To recap, here are the steps we followed to find the inverse function:

1. Replace f(x) with y.
2. Swap x and y.
3. Isolate y (this often involves several algebraic steps).
4. Replace y with f⁻¹(x).

By following these steps, you can find the inverse of many different functions. Just remember to pay close attention to the order of operations and be careful with your algebra!

Verification (Optional but Recommended)

To be absolutely sure we've found the correct inverse, we can verify our answer. The key to verification lies in the fundamental property of inverse functions we discussed earlier: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. In other words, if we plug the inverse function into the original function (or vice versa), we should get x as the result.

Let's try verifying with f(f⁻¹(x)):

This involves plugging the entire expression for f⁻¹(x) into f(x), which can look a bit messy, but it's a worthwhile check.

f(f⁻¹(x)) = 2∛(3 * [((x + 5) / 2)³ + 1] / 3 - 1) - 5

Now, we need to simplify this expression carefully:

1. The 3's inside the cube root cancel out:

   f(f⁻¹(x)) = 2∛(((x + 5) / 2)³ + 1 - 1) - 5

2. The 1's cancel out:

   f(f⁻¹(x)) = 2∛(((x + 5) / 2)³) - 5

3. The cube root and the cube cancel each other out:

   f(f⁻¹(x)) = 2 * ((x + 5) / 2) - 5

4. The 2's cancel out:

   f(f⁻¹(x)) = (x + 5) - 5

5. The 5's cancel out:

   f(f⁻¹(x)) = x

As we can see, f(f⁻¹(x)) = x, which confirms that we have indeed found the correct inverse function! You can also verify f⁻¹(f(x)) = x using the same method. It involves a bit more algebra, but it will give you further confidence in your answer.

Conclusion

So there you have it! We've successfully navigated the process of finding the inverse of a somewhat complex function. Remember, the key is to understand the concept of inverse functions, follow the steps carefully, and pay attention to detail in your algebraic manipulations. Practice makes perfect, so try finding the inverses of other functions to solidify your understanding. You've got this!