To find $(f \circ g)(x)$, also known as $f(g(x))$, we substitute $g(x)$ into $f(x)$.

Given:

• $f(x) = x + 2$
• $g(x) = 2x^2 + 3x + 6$

Steps:

1. Substitute $g(x)$ into $f(x)$: $f(g(x)) = g(x) + 2$

2. Replace $g(x)$ with $2x^2 + 3x + 6$: $f(g(x)) = (2x^2 + 3x + 6) + 2$

3. Simplify the expression: $f(g(x)) = 2x^2 + 3x + 6 + 2$

4. Combine like terms: $f(g(x)) = 2x^2 + 3x + 8$

Final Answer:

$(f \circ g)(x) = 2x^2 + 3x + 8$

Would you like a detailed breakdown of this process or additional examples?

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Related Questions:

1. How do you find $(g \circ f)(x)$ using the given functions?
2. What is the domain of $(f \circ g)(x)$ for these specific functions?
3. How does the composition $(f \circ g)(x)$ differ when $f(x)$ or $g(x)$ changes?
4. What is the inverse of $(f \circ g)(x)$, if it exists?
5. Can you verify $(f \circ g)(x)$ graphically by plotting $f(x)$, $g(x)$, and their composition?

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Tip:

Always simplify your final expression after substituting one function into another to ensure the polynomial is in its simplest form.