Calculating $f^{-1}(f(-12))+3f(f(18))$: A Step-by-Step Guide

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Hey guys! Today, we're diving into a fun little math problem that involves function evaluation and inverse functions. We're going to break down how to calculate the value of $f^{-1}(f(-12))+3f(f(18))$ using a given table of values for the function $f(x)$. This might sound intimidating, but trust me, we'll take it one step at a time and make it super clear. So, let's jump right in and get started!

Understanding the Problem

Before we get into the nitty-gritty calculations, let's make sure we understand what the problem is asking. We're given a function $f(x)$ and a table of its values for certain inputs. Our mission, should we choose to accept it (and we do!), is to find the value of the expression $f^{-1}(f(-12))+3f(f(18))$.

To tackle this, we need to understand a few key concepts:

• Function Evaluation: This simply means plugging a value into a function and finding the output. For example, if we know $f(2) = 5$, we've evaluated the function $f$ at $x = 2$.
• Composite Functions: These are functions within functions, like $f(f(18))$. It means we first evaluate the inner function ($f(18)$ in this case) and then use that result as the input for the outer function.
• Inverse Functions: The inverse function, denoted as $f^{-1}(x)$, "undoes" what the original function does. If $f(a) = b$, then $f^{-1}(b) = a$. Think of it as going backwards through the function.

With these concepts in mind, we're ready to roll up our sleeves and get calculating!

Step-by-Step Calculation

Let's break down the expression $f^{-1}(f(-12))+3f(f(18))$ into smaller, manageable parts.

1. Evaluating $f(-12)$

The first thing we need to do is find the value of $f(-12)$. To do this, we'll look at the table provided. The table gives us pairs of $x$ and $f(x)$ values. We need to find the row where $x = -12$ and see what the corresponding $f(x)$ value is.

According to the table:

$x$	$f(x)$
10	2
-18	18
7	-3
18	10
-12	-8
0	-1

We can see that when $x = -12$, $f(x) = -8$. So, we have:

$f(-12) = -8$

This is a crucial first step because it simplifies the first part of our expression. Now we know we need to find $f^{-1}(-8)$.

2. Evaluating $f^{-1}(f(-12)) = f^{-1}(-8)$

Next up, we need to figure out what $f^{-1}(-8)$ is. Remember that the inverse function undoes the original function. So, $f^{-1}(-8)$ is asking the question: "What value of $x$ makes $f(x) = -8$?".

To find this, we'll look at our table again, but this time we're looking for the $f(x)$ value that equals -8. Once we find it, we'll note the corresponding $x$ value.

Looking back at the table:

$x$	$f(x)$
10	2
-18	18
7	-3
18	10
-12	-8
0	-1

We can see that when $f(x) = -8$, the corresponding $x$ value is -12. Therefore:

$f^{-1}(-8) = -12$

So, the first part of our expression, $f^{-1}(f(-12))$, simplifies to -12. Awesome! Let's move on to the second part.

3. Evaluating $f(18)$

Now we need to tackle the second part of the expression: $3f(f(18))$. This involves a composite function, so we'll start with the inner function, $f(18)$.

Again, we'll consult our trusty table. This time, we're looking for the row where $x = 18$ and noting the corresponding $f(x)$ value.

From the table:

$x$	$f(x)$
10	2
-18	18
7	-3
18	10
-12	-8
0	-1

We find that when $x = 18$, $f(x) = 10$. So:

$f(18) = 10$

Now we know that we need to find $f(10)$.

4. Evaluating $f(f(18)) = f(10)$

Now that we know $f(18) = 10$, we need to find $f(10)$. This means we're evaluating the function $f$ at $x = 10$.

Let's go back to the table one more time:

$x$	$f(x)$
10	2
-18	18
7	-3
18	10
-12	-8
0	-1

Looking at the table, when $x = 10$, $f(x) = 2$. Therefore:

$f(10) = 2$

So, $f(f(18)) = 2$.

5. Evaluating $3f(f(18))$

We're almost there! We've found that $f(f(18)) = 2$, but we need to find $3f(f(18))$. This is a simple multiplication:

$3f(f(18)) = 3 * 2 = 6$

6. Putting it All Together

Finally, we have all the pieces we need to solve the original expression:

$f^{-1}(f(-12))+3f(f(18))$

We found that:

• $f^{-1}(f(-12)) = -12$
• $3f(f(18)) = 6$

So, we can substitute these values back into the expression:

$f^{-1}(f(-12))+3f(f(18)) = -12 + 6$

And now, the final calculation:

$-12 + 6 = -6$

Final Answer

Therefore, $f^{-1}(f(-12))+3f(f(18)) = -6$.

Key Takeaways

Alright, guys, we did it! We successfully navigated through this function evaluation problem. Let's recap the main steps we took:

1. Understand the Problem: We made sure we knew what the question was asking and the concepts involved (function evaluation, composite functions, and inverse functions).
2. Break it Down: We broke the complex expression into smaller, more manageable parts.
3. Use the Table: We used the table of values to find the function values and inverse function values we needed.
4. Step-by-Step Calculation: We carefully calculated each part of the expression, one at a time.
5. Put it All Together: We combined the results of our calculations to find the final answer.

This problem highlights the importance of understanding the definitions of functions and inverse functions. By breaking down the problem into smaller steps and using the given table effectively, we were able to arrive at the correct solution. Keep practicing, and you'll become a pro at these types of problems in no time!

Practice Problems

Want to test your skills? Here are a couple of practice problems you can try:

1. Using the same table, find the value of $2f^{-1}(f(18)) - f(f(-12))$.

2. Suppose $g(x)$ is a function with the following values:

   $x$	$g(x)$
   -2	5
   0	1
   1	-2
   3	0
   5	3

   Find the value of $g^{-1}(g(1)) + 4g(g(0))$.

Try these out and see if you can apply the same techniques we used in this article. Happy calculating!

Conclusion

I hope this step-by-step guide has helped you understand how to calculate expressions involving function evaluation and inverse functions. Remember, math can be fun, especially when you break it down and tackle it one step at a time. Keep practicing, keep exploring, and most importantly, keep enjoying the process! If you have any questions or want to dive deeper into similar problems, feel free to ask. Until next time, happy math-ing!