Finding The Inverse: Solving For F⁻¹(2) | Math Problem

Hey guys! Let's dive into a fun math problem today where we're going to find the inverse of a function and then evaluate it at a specific point. This is a classic type of question you might see in algebra or pre-calculus, and it's super important to understand how to tackle these. So, let's get started!

Understanding the Problem

First, let’s break down what we're given. We have a function f(x) = rac{9x+8}{x-3}, and we know that $x$ cannot be equal to 3 because that would make the denominator zero, which is a big no-no in math. We're also told that $f^{-1}(x)$ is the inverse of our function $f(x)$. Our mission, should we choose to accept it (and we do!), is to find the value of $f^{-1}(2)$. That means we need to figure out what input value for $f(x)$ would give us an output of 2. Understanding the concept of inverse functions is crucial here. An inverse function essentially "undoes" what the original function does. If $f(a) = b$, then $f^{-1}(b) = a$. This relationship is the key to solving our problem.

To further clarify, think of it this way: the original function $f(x)$ takes an input $x$, performs some operations on it (in this case, multiplying by 9, adding 8, and then dividing by $x-3$), and gives us an output. The inverse function $f^{-1}(x)$ takes an output from the original function as its input and returns the original input $x$. So, finding $f^{-1}(2)$ means we're looking for the value of $x$ that, when plugged into $f(x)$, gives us 2. This understanding is fundamental to the steps we’ll take to solve the problem. We’re not just plugging numbers into a formula; we're thinking about the relationship between the function and its inverse. The restriction $x eq 3$ is also important because it tells us the domain of the function $f(x)$ and, consequently, the range of the inverse function $f^{-1}(x)$.

Finding the Inverse Function

Okay, so the first big step is to actually find the inverse function, $f^{-1}(x)$. There's a pretty standard method for doing this, and once you've done it a few times, it becomes second nature. Here’s how we do it:

1. Replace $f(x)$ with $y$: This makes the equation a little easier to work with. So, we rewrite f(x) = rac{9x+8}{x-3} as y = rac{9x+8}{x-3}. This is just a notational change, but it sets us up for the next steps.
2. Swap $x$ and $y$: This is the heart of finding the inverse. We're essentially reversing the roles of input and output. So, we get x = rac{9y+8}{y-3}. This step reflects the fundamental idea of an inverse function: it reverses the mapping from $x$ to $y$.
3. Solve for $y$: Now, we need to isolate $y$ on one side of the equation. This will give us the equation for the inverse function. Let's walk through the algebra:
   • Multiply both sides by $(y-3)$: This gets rid of the fraction. We have $x(y-3) = 9y + 8$.
   • Distribute the $x$: We get $xy - 3x = 9y + 8$.
   • Get all the $y$ terms on one side and the non-$y$ terms on the other: We can do this by subtracting $9y$ from both sides and adding $3x$ to both sides. This gives us $xy - 9y = 3x + 8$.
   • Factor out $y$: We have $y(x - 9) = 3x + 8$.
   • Divide both sides by $(x-9)$: Finally, we isolate $y$ and get y = rac{3x+8}{x-9}.
4. Replace $y$ with $f^{-1}(x)$: This is the final step to express our answer in the correct notation. So, we have f^{-1}(x) = rac{3x+8}{x-9}. This is our inverse function! We’ve successfully found the function that “undoes” the original function $f(x)$.

Each of these steps is crucial, and understanding the reasoning behind them makes the process much clearer. Swapping $x$ and $y$ is the key conceptual step, and the algebra that follows is simply a matter of manipulating the equation to isolate $y$. Make sure you’re comfortable with these algebraic manipulations, as they’re a common skill needed in many math problems.

Evaluating the Inverse Function

Alright, now that we've found the inverse function, f^{-1}(x) = rac{3x+8}{x-9}, we're ready for the final part of the problem: finding the value of $f^{-1}(2)$. This is actually the easy part now that we have the inverse function. All we need to do is substitute 2 for $x$ in the expression for $f^{-1}(x)$. Let's do it!

So, we have:

f^{-1}(2) = rac{3(2)+8}{2-9}

Now, let's simplify:

f^{-1}(2) = rac{6+8}{2-9} = rac{14}{-7} = -2

And there you have it! We've found that $f^{-1}(2) = -2$. So, the value of the inverse function at $x = 2$ is -2. This means that if we plug -2 into the original function $f(x)$, we should get 2 as the output. You could even check this by plugging -2 into $f(x)$ to verify your answer.

This process of evaluating the inverse function is straightforward once you have the expression for the inverse function itself. It’s a simple matter of substitution and arithmetic. However, it’s crucial to remember that this step relies on having correctly found the inverse function in the first place. So, double-checking your work in the previous steps is always a good idea.

The Answer

So, after all that work, we've arrived at our answer. The value of $f^{-1}(2)$ is -2. Looking at the options provided, we can see that this corresponds to option (A).

Key Takeaways

This problem is a great example of how to work with inverse functions. Here are some key things to remember:

• What is an inverse function? An inverse function "undoes" the original function. If $f(a) = b$, then $f^{-1}(b) = a$.
• How to find the inverse function:
  1. Replace $f(x)$ with $y$.
  2. Swap $x$ and $y$.
  3. Solve for $y$.
  4. Replace $y$ with $f^{-1}(x)$.
• How to evaluate the inverse function: Once you have the expression for $f^{-1}(x)$, simply substitute the given value for $x$ and simplify.

Understanding these concepts and steps will help you tackle a wide range of problems involving inverse functions. Practice is key, so try working through similar examples to solidify your understanding. And remember, if you ever get stuck, break the problem down into smaller steps and tackle each one individually.

Practice Problems

Want to test your understanding? Try these practice problems:

1. Given g(x) = rac{5x-1}{x+2}, $x eq -2$, find $g^{-1}(x)$ and evaluate $g^{-1}(3)$.
2. If $h(x) = 2x^3 + 7$, find $h^{-1}(x)$.

Working through these problems will give you valuable practice and help you build confidence in your ability to work with inverse functions. Remember, math is a skill that improves with practice, so don’t be afraid to challenge yourself!

Conclusion

So, guys, we've successfully solved for $f^{-1}(2)$! We walked through the steps of finding the inverse function and then evaluating it at a specific point. Remember, understanding the concepts behind the steps is just as important as being able to perform the calculations. Keep practicing, and you'll become a pro at working with inverse functions in no time! Keep up the great work, and I'll see you in the next math adventure! Remember to always double-check your work and understand the logic behind each step. Math is not just about getting the right answer; it's about understanding the process. Happy problem-solving!