Express your answer as a polynomial in standard form.
f, of, x, equals, 3, x, minus, 4
f(x)=
3x−4
g, of, x, equals, x, squared, plus, 6, x, minus, 4
g(x)=
x 
2
 +6x−4
Find: , g, of, f, of, x
Find: g(f(x))

We are asked to find \( g(f(x)) \), which means we need to substitute \( f(x) = 3x - 4 \) into the function \( g(x) = x^2 + 6x - 4 \).

### Step 1: Substitution
Start by replacing every occurrence of \( x \) in \( g(x) \) with \( f(x) = 3x - 4 \):
\[
g(f(x)) = (3x - 4)^2 + 6(3x - 4) - 4
\]

### Step 2: Expand the expression
First, expand \( (3x - 4)^2 \):
\[
(3x - 4)^2 = (3x)^2 - 2(3x)(4) + 4^2 = 9x^2 - 24x + 16
\]

Next, expand \( 6(3x - 4) \):
\[
6(3x - 4) = 18x - 24
\]

### Step 3: Combine all terms
Now substitute these expanded expressions into the equation for \( g(f(x)) \):
\[
g(f(x)) = 9x^2 - 24x + 16 + 18x - 24 - 4
\]

### Step 4: Simplify
Now, combine like terms:
\[
g(f(x)) = 9x^2 + (-24x + 18x) + (16 - 24 - 4)
\]
\[
g(f(x)) = 9x^2 - 6x - 12
\]

### Final Answer:
The polynomial in standard form is:
\[
g(f(x)) = 9x^2 - 6x - 12
\]

Would you like more details on any step?

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Here are 5 related questions to expand on this:

1. What is the general process for finding \( f(g(x)) \) instead of \( g(f(x)) \)?
2. How do you find the degree of a polynomial after substitution?
3. Can you apply this method to any function combination, such as \( h(f(x)) \)?
4. What are the advantages of expressing polynomials in standard form?
5. How does simplifying complex expressions help in calculus?

**Tip:** When expanding binomials, use the FOIL method to avoid errors!

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 guide explains how to solve for g(f(x)) where f(x) = 3x - 4 and g(x) = x^2 + 6x - 4. Learn how to substitute and expand expressions step by step to arrive at the polynomial in standard form, 9x^2 - 6x - 12. Ideal for students in Grades 9-11.

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