1. Define the rule for a function $f$ that determines $y$ in terms of $x$:

The given equation is:

$y = 3x + 13$

This can be expressed as a function $f$ where $y = f(x)$:

$f(x) = 3x + 13$

2. Solve the equation $y = 3x + 13$ for $x$:

To find $x$ in terms of $y$, we solve for $x$:

$y = 3x + 13$ $y - 13 = 3x$ $x = \frac{y - 13}{3}$

Thus,

$x = \frac{y - 13}{3}$

3. Define the rule for a function $g$ that determines $x$ in terms of $y$:

The function $g(y)$ is the inverse of $f(x)$, which determines $x$ from $y$. From the solution above:

$g(y) = \frac{y - 13}{3}$

4. Which of the following are true?

• $f$ and $g$ are inverses: This is true because $f(x)$ transforms $x$ to $y$, and $g(y)$ reverses this transformation back to $x$.
• $g = f^{-1}$: This is true, as $g(y)$ undoes the effect of $f(x)$, so $g$ is the inverse of $f$.
• $f = g^{-1}$: This is true since $f(x)$ is the inverse of $g(y)$, reversing $g$'s process.
• $g$ undoes the process of $f$: This is true because $g$ is the inverse of $f$, and undoing the process is what an inverse function does.

Thus, all statements are true.

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Do you want more details or have any questions?

Here are 5 related questions to consider:

1. How do you verify that two functions are inverses of each other?
2. Can all linear functions have inverses? Why or why not?
3. What is the graphical relationship between a function and its inverse?
4. How do inverses of functions behave in composite function operations?
5. What would happen if you shifted the function $f(x) = 3x + 13$?

Tip:

When solving for the inverse of a function, always check by composing the original function and the inverse to ensure $f(g(y)) = y$ and $g(f(x)) = x$.