If f(x) = 3x-5 and g(f(x)) = 9x^2 -30x - 5, find g(x)

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:
\[
g(x) = x^2 - 30
\]

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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.

Learn how to solve a function composition problem by finding g(x) when f(x) = 3x - 5 and g(f(x)) = 9x^2 - 30x - 5. This solution involves algebraic manipulation, quadratic expansion, and function decomposition, suitable for students in Grades 9-11.

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