To find the function $g(x)$, we are given that:

$(g \circ f)(x) = g(f(x)) = 2x^2 + 6x - 7$ $f(x) = x^2 + 3x - 4$

This means that $g(f(x)) = 2x^2 + 6x - 7$, and we need to express $g(x)$ such that when we substitute $f(x)$ into $g(x)$, we get $2x^2 + 6x - 7$.

Let's assume that $g(x)$ is of the form $g(x) = ax + b$. To solve this, we need to find $a$ and $b$.

Substitute $f(x)$ into $g(x)$:

$g(f(x)) = a(f(x)) + b = a(x^2 + 3x - 4) + b$

Now expand:

$g(f(x)) = a(x^2 + 3x - 4) + b = ax^2 + 3ax - 4a + b$

We know that $g(f(x)) = 2x^2 + 6x - 7$, so we can compare coefficients:

$ax^2 + 3ax - 4a + b = 2x^2 + 6x - 7$

From the comparison of coefficients, we get:

1. $a = 2$ (from the $x^2$ terms)
2. $3a = 6$ which is also true since $3 \times 2 = 6$ (from the $x$ terms)
3. $-4a + b = -7$

Substitute $a = 2$ into the third equation:

$-4(2) + b = -7 \implies -8 + b = -7 \implies b = 1$

Thus, $g(x) = 2x + 1$.

Final Answer:

$g(x) = 2x + 1$

Would you like further details or explanations on any part of this?

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Here are 5 related questions:

1. How do you compose functions in general?
2. What happens if $f(x)$ is a higher-degree polynomial?
3. How can you verify if two functions are inverses of each other?
4. How do you solve for unknowns in function compositions?
5. What other forms can a composite function take?

Tip: When comparing functions, always align terms by their degree (constant, linear, quadratic, etc.) to easily match coefficients.