To find $f^{-1}(f(2)) - 3f(f(-2))$, we need to evaluate each component step by step, starting with $f(x)$.

Given the function values:

• $f(2) = 22$
• $f(0) = 0$
• $f(-2) = -6$
• $f(1) = -2$
• $f(10) = 22$

Step 1: Find $f^{-1}(f(2))$

Since $f(2) = 22$, we need to find $f^{-1}(22)$. Observing the provided values, we see:

• $f(10) = 22$

Thus, $f^{-1}(22) = 10$.

Step 2: Calculate $f(f(-2))$

Next, we need to evaluate $f(f(-2))$. From the values:

• $f(-2) = -6$

Now we evaluate $f(-6)$. However, we don’t have $f(-6)$ listed in the values provided, so we cannot determine this part without additional information. Assuming $f(-6)$ is defined, we denote it as $f(-6) = a$.

Step 3: Substitute into the expression

Now we can rewrite the expression:

$f^{-1}(f(2)) - 3f(f(-2)) = 10 - 3f(-6) = 10 - 3a$

Conclusion

Without the value of $f(-6)$, we cannot simplify further. The final expression is:

$10 - 3a$

Summary

1. Result: $f^{-1}(f(2)) - 3f(f(-2)) = 10 - 3f(-6)$
2. Known values: $f(2) = 22$, $f(-2) = -6$, $f(10) = 22$

Do you have any questions about this solution, or is there additional information about $f(-6)$?

Additional Questions

1. What is the value of $f(-6)$?
2. Are there any more function values provided?
3. Would you like help defining $f(x)$ if it’s a specific function?
4. How does $f^{-1}(x)$ relate to the function values given?
5. Do you want to explore the implications of inverse functions in more detail?

Tip

Always ensure to provide enough function values when dealing with inverses, as missing values can lead to incomplete results.