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### 15. Assume that \( f \) is a one-to-one function.
#### (a) If \( f(6) = 17 \), what is \( f^{-1}(17) \)?

- Since \( f \) is one-to-one, for each output there is exactly one input.
- If \( f(6) = 17 \), it means that 6 is mapped to 17 by the function \( f \).
- The inverse function \( f^{-1} \) does the reverse: it maps 17 back to 6.

So, \( f^{-1}(17) = 6 \).

#### (b) If \( f^{-1}(3) = 2 \), what is \( f(2) \)?

- By the definition of the inverse function, \( f^{-1}(3) = 2 \) means that \( f(2) = 3 \).

So, \( f(2) = 3 \).

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### 16. If \( f(x) = x^3 + x^3 + x \), find \( f^{-1}(3) \) and \( f(f^{-1}(2)) \).

#### Step 1: Simplify the function
The function is given as:
\[
f(x) = x^3 + x^3 + x = 2x^3 + x
\]

#### (a) Finding \( f^{-1}(3) \):
We are asked to find \( f^{-1}(3) \), which means we need to solve for \( x \) such that:
\[
f(x) = 3 \implies 2x^3 + x = 3
\]

This is a cubic equation, and solving for \( x \) requires factoring or applying numerical methods.

#### (b) Finding \( f(f^{-1}(2)) \):
By the definition of the inverse function:
\[
f(f^{-1}(2)) = 2
\]

This is because applying a function and then its inverse always brings us back to the original input.

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Do you want any further details on solving the cubic equation for part (a) of question 16? 

Here are 5 related questions that expand on this information:

1. How do you generally solve cubic equations?
2. What is the graphical interpretation of the inverse of a function?
3. Can all functions have inverses? Why or why not?
4. How does the concept of one-to-one functions relate to their inverses?
5. What numerical methods can be used to solve complex cubic equations?

**Tip:** When working with inverse functions, always remember that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \), as long as the inverse exists.

15. Assume that f is a one-to-one function. (a) If f(6) = 17, what is f^{-1}(17)? (b) If f^{-1}(3) = 2, what is f(2)? 16. If f(x) = x^3 + x^3 + x, find f^{-1}(3) and f(f^{-1}(2)).

Explore how to find inverse functions for one-to-one mappings and solve cubic equations. This problem includes concepts like f(f^{-1}(x)) = x and solving equations of the form f(x) = 2x^3 + x. Suitable for college-level students.

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