To find $(f \circ g)(x)$, or $f(g(x))$, we substitute $g(x)$ into $f(x)$. Here are the given functions:

• $f(x) = 3x^2$
• $g(x) = x + 2$

Steps:

1. Replace $x$ in $f(x)$ with $g(x)$: $f(g(x)) = f(x + 2)$

2. Substitute $x + 2$ into $f(x) = 3x^2$: $f(x + 2) = 3(x + 2)^2$

3. Expand $(x + 2)^2$: $(x + 2)^2 = x^2 + 4x + 4$

4. Multiply the expanded expression by 3: $3(x^2 + 4x + 4) = 3x^2 + 12x + 12$

Final Answer:

$(f \circ g)(x) = 3x^2 + 12x + 12$

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Related Questions:

1. How do you find $(g \circ f)(x)$ for the same functions?
2. What is $(f \circ g)(2)$ using the derived expression?
3. How do you verify $(f \circ g)(x)$ algebraically?
4. What happens if $g(x) = x^2 - 1$ instead? Find $(f \circ g)(x)$.
5. How do you graph $(f \circ g)(x)$ alongside $f(x)$ and $g(x)$?

Tip:

When solving composite functions, always substitute carefully to ensure correct replacements and simplifications.