Let's analyze the problem step by step.

We are given two functions:

• $f(x) = 3x - 2$
• $(g \circ f)(x) = 9x^2 - 7$, which means $g(f(x)) = 9x^2 - 7$

We need to find the formula for $g(x)$.

Step-by-Step Solution:

1. First, substitute $f(x) = 3x - 2$ into the composition $g(f(x))$: $g(3x - 2) = 9x^2 - 7$

2. To find $g(x)$, let's set $u = 3x - 2$, so the equation becomes: $g(u) = 9x^2 - 7$

3. Solve for $x$ in terms of $u$: $u = 3x - 2 \quad \Rightarrow \quad 3x = u + 2 \quad \Rightarrow \quad x = \frac{u + 2}{3}$

4. Now substitute $x = \frac{u + 2}{3}$ back into $9x^2 - 7$: $9x^2 = 9\left(\frac{u + 2}{3}\right)^2 = 9 \times \frac{(u + 2)^2}{9} = (u + 2)^2$ So: $g(u) = (u + 2)^2 - 7$ Expand $(u + 2)^2$: $g(u) = u^2 + 4u + 4 - 7 = u^2 + 4u - 3$

5. Therefore, the function $g(x)$ is: $g(x) = x^2 + 4x - 3$

The correct answer is Option D.

Would you like any more details or explanations?

Here are some related questions for further practice:

1. If $f(x) = 2x + 1$ and $g(f(x)) = 4x^2 + 5x$, find $g(x)$.
2. How would you verify if $g(x) = x^2 + 4x - 3$ is correct for all $x$?
3. How do we derive the inverse of $f(x) = 3x - 2$?
4. If $(g \circ h)(x) = 25x^2 - 16$, and $h(x) = 5x + 4$, find $g(x)$.
5. What is the domain and range of the function $g(x) = x^2 + 4x - 3$?

Tip: When working with function compositions, always substitute the given functions carefully and check for algebraic simplifications.