g°f) (x) = 2x² + 6x -7 f(x)= x² + 3x - 4 g(x)=?

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.

Solve a function composition problem where g(f(x)) = 2x² + 6x -7 and f(x) = x² + 3x - 4 to find g(x). This detailed step-by-step solution explores key concepts in algebra and function composition, suitable for students in Grades 9-12.

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