We are given two functions:

1. $f(x) = 3x - 5$
2. $g(f(x)) = 9x^2 - 30x - 5$

We are tasked with finding $g(x)$.

Step 1: Express $f(x)$ in terms of $g(x)$

From the problem, we know that $g(f(x)) = 9x^2 - 30x - 5$. This suggests that $g$ is a function of the form $g(f(x)) = g(3x - 5)$.

Step 2: Substitute $f(x) = 3x - 5$ into the equation

Let’s denote $y = 3x - 5$, meaning that $g(y) = 9x^2 - 30x - 5$.

Now, we need to express $x$ in terms of $y$. Solve for $x$ from $y = 3x - 5$:

$y = 3x - 5$ $y + 5 = 3x$ $x = \frac{y + 5}{3}$

Step 3: Rewrite $g(y)$

Now, substitute $x = \frac{y + 5}{3}$ into the expression for $g(f(x))$:

$g(y) = 9\left( \frac{y + 5}{3} \right)^2 - 30\left( \frac{y + 5}{3} \right) - 5$

Step 4: Simplify the expression for $g(y)$

First, simplify the square term: $\left( \frac{y + 5}{3} \right)^2 = \frac{(y + 5)^2}{9}$ Thus: $g(y) = 9 \cdot \frac{(y + 5)^2}{9} - 30 \cdot \frac{y + 5}{3} - 5$

Simplifying further: $g(y) = (y + 5)^2 - 10(y + 5) - 5$

Now, expand the terms: $g(y) = (y^2 + 10y + 25) - 10(y + 5) - 5$ $g(y) = y^2 + 10y + 25 - 10y - 50 - 5$ $g(y) = y^2 - 30$

Thus, the function $g(x)$ is:

$g(x) = x^2 - 30$

Final Answer:

$g(x) = x^2 - 30$

---

Do you want more details on this solution, or do you have any questions?

Here are 5 related questions to expand on this:

1. How do you verify if two functions are inverses of each other?
2. Can you find the composition $f(g(x))$ given $f(x) = 3x - 5$ and $g(x) = x^2 - 30$?
3. What happens to the graph of $f(x)$ when you compose it with another function?
4. How do you find the domain and range of $g(x)$?
5. What are the steps to derive the inverse of a function?

Tip: When solving composition problems, carefully track substitutions and ensure to simplify step by step.