Let
f
(
x
)
=
2
x
+
2
and
g
(
x
)
=
3
x
2
+
3
x
.
After simplifying,
(
f
∘
g
)
(
x
)

Given the functions:

\( f(x) = 2x + 2 \) and \( g(x) = 3x^2 + 3x \),

we want to find \( (f \circ g)(x) \), which means applying \( f \) to the result of \( g(x) \). This can be written as:

\[
(f \circ g)(x) = f(g(x)).
\]

### Step 1: Substitute \( g(x) \) into \( f(x) \)
Since \( g(x) = 3x^2 + 3x \), we substitute this into \( f(x) \):

\[
f(g(x)) = f(3x^2 + 3x).
\]

Now, using the definition of \( f(x) \), which is \( f(x) = 2x + 2 \), we replace \( x \) in \( f(x) \) with \( 3x^2 + 3x \):

\[
f(3x^2 + 3x) = 2(3x^2 + 3x) + 2.
\]

### Step 2: Simplify the expression
Distribute the 2:

\[
f(3x^2 + 3x) = 2 \cdot 3x^2 + 2 \cdot 3x + 2.
\]

This gives:

\[
f(3x^2 + 3x) = 6x^2 + 6x + 2.
\]

Thus, the simplified form of \( (f \circ g)(x) \) is:

\[
(f \circ g)(x) = 6x^2 + 6x + 2.
\]

---

Do you need further clarification or details on any step?

Here are 5 follow-up questions to consider:
1. How would the result change if we switch the composition to \( g \circ f \)?
2. What happens if we differentiate \( (f \circ g)(x) \)?
3. Can we compose more functions in a chain, like \( h(x) \circ f(x) \circ g(x) \)?
4. How would the composition \( (f \circ g)(x) \) behave for negative values of \( x \)?
5. What are the general properties of function composition?

**Tip**: When composing functions, always substitute one function's output into the other carefully and simplify step by step.

Let f(x) = 2x + 2 and g(x) = 3x^2 + 3x. After simplifying, (f ∘ g)(x).

Learn how to simplify the composition of two functions, f(x) = 2x + 2 and g(x) = 3x^2 + 3x, to find (f ∘ g)(x). This step-by-step guide covers key algebraic concepts and function composition rules, suitable for students in Grades 9-12.

Math Problem Statement

Let f ( x )

2 x + 2 and g ( x )

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