Let's go through each problem visible in the image step by step.

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, $f(2) = 3$.

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.

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.