Finding The Inverse Of G(x) = -2/3x - 5 A Step-by-Step Guide

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Hey guys! Ever wondered how to undo a function? That's where the concept of an inverse function comes in. Think of it like this: if a function is a machine that takes an input and spits out an output, the inverse function is the machine that takes that output and spits back the original input. Cool, right? In this article, we're diving deep into finding the inverse of a specific function: g(x) = -2/3x - 5. We'll break it down step by step, so even if you're just starting your journey into the world of functions, you'll be able to follow along. So, let's get started and unlock the mystery of inverse functions!

Understanding Inverse Functions

Before we jump into the nitty-gritty of finding the inverse of g(x) = -2/3x - 5, let's make sure we're all on the same page about what an inverse function actually is. In simple terms, the inverse function reverses the action of the original function. Imagine you have a function that doubles a number and then adds 3. The inverse function would have to undo those operations, first subtracting 3 and then halving the result. This "undoing" process is the core idea behind inverse functions.

Key Idea: If we apply a function and then its inverse, we should end up back where we started. Mathematically, this can be expressed as follows:

  • If f(x) is a function and f⁻¹(x) is its inverse, then:
    • f⁻¹(f(x)) = x
    • f(f⁻¹(x)) = x

This might seem a bit abstract, but it's a powerful concept. It basically says that if you plug a value into a function, get the output, and then plug that output into the inverse function, you'll get back your original value. Let's illustrate this with a simple example before tackling our main function, g(x) = -2/3x - 5. Consider the function f(x) = x + 2. The inverse function would be f⁻¹(x) = x - 2. If we start with x = 3, then f(3) = 3 + 2 = 5. Now, if we plug 5 into the inverse function, we get f⁻¹(5) = 5 - 2 = 3, which is our original value! See how the inverse function "undid" the original function?

Now, a crucial thing to remember is that not every function has an inverse. For a function to have an inverse, it must be one-to-one. This means that each input value (x) corresponds to a unique output value (y), and vice versa. Graphically, a one-to-one function passes the horizontal line test, meaning that no horizontal line intersects the graph more than once. If a function fails the horizontal line test, it doesn't have a true inverse over its entire domain.

So, before we even try to find the inverse of g(x) = -2/3x - 5, it's a good idea to check if it's a one-to-one function. Linear functions, like the one we're dealing with, are generally one-to-one (except for horizontal lines, which are constant functions). Since our function has a non-zero slope (-2/3), we can be confident that it's one-to-one and has an inverse.

Now that we've got a solid grasp of what inverse functions are and how they work, let's move on to the step-by-step process of finding the inverse of g(x) = -2/3x - 5. We'll take it nice and slow, so you'll be a pro in no time!

Step 1: Replace g(x) with y

Okay, let's get our hands dirty with the actual process of finding the inverse of our function, g(x) = -2/3x - 5. The first step is a simple one, but it's crucial for making the algebra a bit easier to handle. We're going to replace the function notation g(x) with the variable y. Remember, g(x) is just a way of representing the output of the function for a given input x, and y is the standard variable we use to represent the dependent variable (the output). So, this substitution is perfectly valid and doesn't change the function itself.

Why do we do this? Well, it's mostly for visual clarity. When we have y instead of g(x), it makes the next steps, which involve swapping variables, feel a bit more natural and less cluttered. Think of it as a cosmetic change that helps us focus on the underlying mathematical operations.

So, applying this first step to our function, g(x) = -2/3x - 5, we simply replace g(x) with y, resulting in the following equation:

y = -2/3x - 5

That's it! Step one is complete. See? Nothing too scary so far. This seemingly small step sets the stage for the next, more crucial step: swapping the x and y variables. This is where the real magic of finding inverse functions happens. But before we jump into that, let's quickly recap why we did this. We replaced g(x) with y to make the equation look simpler and prepare it for the variable swap that's coming up next. This substitution is a common practice when finding inverse functions, and it's a handy trick to keep in your mathematical toolkit.

Now that we've laid the groundwork, let's move on to the heart of the inverse function finding process: swapping x and y. This step might seem a little strange at first, but it's the key to "undoing" the original function and finding its inverse. So, let's dive in and see how it works!

Step 2: Swap x and y

Alright, guys, we've reached the most crucial step in finding the inverse of a function: swapping x and y. This is where the magic happens, where we actually begin to "undo" the original function. Remember, the inverse function reverses the roles of input and output. So, if the original function takes an x-value and produces a y-value, the inverse function should take that y-value and produce the original x-value. This swapping of variables is the mathematical way of expressing that reversal.

Think about it this way: the original function defines y in terms of x. The inverse function, on the other hand, should define x in terms of y. By swapping x and y, we're setting ourselves up to solve for y in terms of x, which will give us the equation for the inverse function.

So, let's apply this step to our equation, which we got from the previous step: y = -2/3x - 5. We simply replace every instance of y with x, and every instance of x with y. This gives us:

x = -2/3y - 5

That's it! We've swapped the variables. Now, we have an equation that expresses x in terms of y. But remember, we want the inverse function to express y in terms of x. So, our next task is to solve this equation for y. This will involve some algebraic manipulation, but don't worry, we'll take it one step at a time.

Before we move on, let's just take a moment to appreciate the significance of this step. Swapping x and y is the cornerstone of finding inverse functions. It's the step that reflects the fundamental idea of reversing the roles of input and output. Without this step, we wouldn't be able to "undo" the original function and find its inverse.

Now that we've swapped the variables, we're ready to move on to the next challenge: solving the equation for y. This will require us to use our algebraic skills to isolate y on one side of the equation. So, let's roll up our sleeves and get to work!

Step 3: Solve for y

Okay, we've swapped x and y in our equation, and now we have x = -2/3y - 5. The next step is to isolate y on one side of the equation. This is a classic algebraic manipulation problem, and we'll use the standard techniques to solve it. Remember, our goal is to get y all by itself on one side, with everything else on the other side.

First, let's get rid of the constant term, -5, on the right side of the equation. To do this, we'll add 5 to both sides. This keeps the equation balanced and moves us closer to isolating y:

x + 5 = -2/3y - 5 + 5

Simplifying, we get:

x + 5 = -2/3y

Now, we need to get rid of the coefficient -2/3 that's multiplying y. The easiest way to do this is to multiply both sides of the equation by the reciprocal of -2/3, which is -3/2. Again, this keeps the equation balanced:

(-3/2)(x + 5) = (-3/2)(-2/3y)

On the right side, the -3/2 and -2/3 cancel each other out, leaving just y. On the left side, we need to distribute the -3/2:

(-3/2)x - (3/2)(5) = y

Simplifying further, we get:

-3/2x - 15/2 = y

We've done it! We've successfully isolated y on one side of the equation. Now we have an equation that expresses y in terms of x. This is the equation for the inverse function! But there's one more step to make it official.

Before we move on, let's recap what we did in this step. We used standard algebraic techniques to isolate y in the equation x = -2/3y - 5. We added 5 to both sides, then multiplied both sides by -3/2. This allowed us to get y by itself and express it in terms of x. This is a crucial skill in finding inverse functions, so make sure you're comfortable with these algebraic manipulations.

Now that we've solved for y, we're ready for the final step: replacing y with the proper inverse function notation. This will give our answer a polished, professional look and clearly indicate that we've found the inverse function.

Step 4: Replace y with g⁻¹(x)

We've done the hard work! We've swapped x and y, solved for y, and now we have the equation y = -3/2x - 15/2. This equation represents the inverse function, but to make it clear that this is the inverse, we need to use the proper notation. That's where this final step comes in. We're going to replace y with g⁻¹(x), which is the standard notation for the inverse of the function g(x). The "-1" in the superscript indicates that this is the inverse function. Think of it as saying, "This is the function that undoes g(x).

So, replacing y with g⁻¹(x) in our equation, we get:

g⁻¹(x) = -3/2x - 15/2

And there you have it! We've found the inverse of the function g(x) = -2/3x - 5. The inverse function is g⁻¹(x) = -3/2x - 15/2. This function will "undo" the operations of the original function. If you plug a value into g(x), get the output, and then plug that output into g⁻¹(x), you should get back your original value. You can try it out with a few examples to verify that it works!

Let's quickly recap this final step. We replaced y with g⁻¹(x) to clearly indicate that we've found the inverse function. This notation is crucial for communicating your result effectively and avoiding any confusion. It tells anyone looking at your work that this function is the inverse of g(x).

Now that we've completed all the steps, let's take a moment to look back at the entire process and appreciate how we found the inverse of g(x) = -2/3x - 5. We started by replacing g(x) with y, then swapped x and y, solved for y, and finally replaced y with g⁻¹(x). By following these steps, we successfully "undid" the original function and found its inverse.

Verification (Optional but Recommended)

Okay, guys, we've found the inverse function, g⁻¹(x) = -3/2x - 15/2, but how can we be absolutely sure that we've done it correctly? Well, there's a handy way to verify our result, and it involves using the key property of inverse functions that we discussed earlier: f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. In other words, if we compose the function with its inverse (in either order), we should get back the original input, x.

Let's verify our result by checking both compositions. First, we'll check g⁻¹(g(x)):

  1. Start with the composition: g⁻¹(g(x))
  2. Substitute g(x) = -2/3x - 5: g⁻¹(-2/3x - 5)
  3. Now, plug -2/3x - 5 into g⁻¹(x) = -3/2x - 15/2: (-3/2)(-2/3x - 5) - 15/2
  4. Distribute the -3/2: (-3/2)(-2/3x) + (-3/2)(-5) - 15/2
  5. Simplify: x + 15/2 - 15/2
  6. The 15/2 terms cancel out, leaving: x

So, g⁻¹(g(x)) = x, which is exactly what we wanted! This confirms that our inverse function works in this direction. Now, let's check the other composition, g(g⁻¹(x)):

  1. Start with the composition: g(g⁻¹(x))
  2. Substitute g⁻¹(x) = -3/2x - 15/2: g(-3/2x - 15/2)
  3. Now, plug -3/2x - 15/2 into g(x) = -2/3x - 5: (-2/3)(-3/2x - 15/2) - 5
  4. Distribute the -2/3: (-2/3)(-3/2x) + (-2/3)(-15/2) - 5
  5. Simplify: x + 5 - 5
  6. The 5 terms cancel out, leaving: x

So, g(g⁻¹(x)) = x as well! This confirms that our inverse function works in both directions. We've successfully verified our result.

Verifying your answer is always a good practice, especially in mathematics. It gives you confidence that you've done the problem correctly and helps you catch any potential errors. In this case, by checking both compositions, we've shown that g⁻¹(x) = -3/2x - 15/2 is indeed the inverse of g(x) = -2/3x - 5.

Conclusion

Alright, we've reached the end of our journey to find the inverse of the function g(x) = -2/3x - 5! We've broken down the process into clear, manageable steps, and we've even verified our result to be absolutely sure. Let's quickly recap the key takeaways:

  • Inverse functions "undo" the original function. If you apply a function and then its inverse, you get back the original input.
  • Not all functions have inverses. A function must be one-to-one (pass the horizontal line test) to have a true inverse.
  • The steps to find the inverse are:
    1. Replace g(x) with y.
    2. Swap x and y.
    3. Solve for y.
    4. Replace y with g⁻¹(x).
  • Verification is crucial! Check your answer by composing the function with its inverse in both directions.

We successfully found that the inverse of g(x) = -2/3x - 5 is g⁻¹(x) = -3/2x - 15/2. This means that if you give g(x) an input, and then give its output to g⁻¹(x), you'll get your original input back. That's the power of inverse functions!

Finding inverse functions might seem a bit tricky at first, but with practice, it becomes a familiar process. The key is to understand the underlying concept of "undoing" the function and to follow the steps carefully. Remember to swap x and y, solve for y, and use the proper notation. And don't forget to verify your answer! It's always better to be safe than sorry.

So, guys, I hope this step-by-step guide has helped you understand how to find the inverse of a function. Now you can go out there and tackle other functions with confidence. Keep practicing, and you'll become a pro at finding inverses in no time! Remember, math is all about understanding the concepts and applying them consistently. And with inverse functions, you've now added another valuable tool to your mathematical toolkit.