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The question asks to find \( g(f(x)) \) and express it as a polynomial in standard form. Here, the functions are:

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

### Steps to solve \( g(f(x)) \):

1. We need to substitute \( f(x) = 3x - 4 \) into the function \( g(x) \), which is defined as \( g(x) = x^2 + 6x - 4 \).
   
2. Replace every instance of \( x \) in \( g(x) \) with \( f(x) = 3x - 4 \). Thus, we get:
   \[
   g(f(x)) = (3x - 4)^2 + 6(3x - 4) - 4
   \]

3. Now, expand the terms:
   - Expand \( (3x - 4)^2 \):
     \[
     (3x - 4)^2 = 9x^2 - 24x + 16
     \]
   - Expand \( 6(3x - 4) \):
     \[
     6(3x - 4) = 18x - 24
     \]

4. Now, combine everything:
   \[
   g(f(x)) = 9x^2 - 24x + 16 + 18x - 24 - 4
   \]

5. Simplify the expression by combining like terms:
   \[
   g(f(x)) = 9x^2 - 6x - 12
   \]

Thus, the polynomial expression for \( g(f(x)) \) is:
\[
g(f(x)) = 9x^2 - 6x - 12
\]

Would you like more details or explanations?

Here are 5 related questions:
1. How do you compose two functions in general?
2. What are the steps to simplify polynomials after expanding?
3. How would the answer change if \( f(x) \) or \( g(x) \) were different?
4. What does it mean to express a function in standard form?
5. How does function composition apply in real-world scenarios?

**Tip:** When composing functions, be careful to substitute the entire function, not just the variable!

Find g(f(x)) where f(x) = 3x - 4 and g(x) = x^2 + 6x - 4. Express your answer as a polynomial in standard form.

This page provides a step-by-step solution to finding g(f(x)), where f(x) = 3x - 4 and g(x) = x^2 + 6x - 4. Learn how to expand and simplify the resulting polynomial, covering key algebraic concepts like function composition and polynomial expansion. Suitable for students in Grades 9-12.

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