To solve these problems, we calculate the composite functions and state their domains. Let's address each part step by step:

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Given:

• $f(x) = x^2$
• $g(x) = x^2 + 3$

The composite functions are defined as follows:

1. $(f \circ g)(x) = f(g(x))$
2. $(g \circ f)(x) = g(f(x))$
3. $(f \circ f)(x) = f(f(x))$
4. $(g \circ g)(x) = g(g(x))$

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(a) $(f \circ g)(x) = f(g(x))$

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

Simplify: $f(g(x)) = x^4 + 6x^2 + 9$

Domain: $g(x) = x^2 + 3$ is a polynomial, so $f(g(x))$ is also defined for all real numbers. $\text{Domain: } (-\infty, \infty)$

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(b) $(g \circ f)(x) = g(f(x))$

Substitute $f(x)$ into $g(x)$: $g(f(x)) = g(x^2) = (x^2)^2 + 3 = x^4 + 3$

Domain: Both $f(x) = x^2$ and $g(x^2) = x^4 + 3$ are polynomials, so $g(f(x))$ is defined for all real numbers. $\text{Domain: } (-\infty, \infty)$

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(c) $(f \circ f)(x) = f(f(x))$

Substitute $f(x)$ into itself: $f(f(x)) = f(x^2) = (x^2)^2 = x^4$

Domain: Since $f(x^2) = x^4$ is a polynomial, it is defined for all real numbers. $\text{Domain: } (-\infty, \infty)$

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(d) $(g \circ g)(x) = g(g(x))$

Substitute $g(x)$ into itself: $g(g(x)) = g(x^2 + 3) = (x^2 + 3)^2 + 3$

Simplify: $g(g(x)) = x^4 + 6x^2 + 9 + 3 = x^4 + 6x^2 + 12$

Domain: Since $g(x^2 + 3)$ is a polynomial, it is defined for all real numbers. $\text{Domain: } (-\infty, \infty)$

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Final Results:

(a) $(f \circ g)(x) = x^4 + 6x^2 + 9$, Domain: $(-\infty, \infty)$
(b) $(g \circ f)(x) = x^4 + 3$, Domain: $(-\infty, \infty)$
(c) $(f \circ f)(x) = x^4$, Domain: $(-\infty, \infty)$
(d) $(g \circ g)(x) = x^4 + 6x^2 + 12$, Domain: $(-\infty, \infty)$

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Do you want a detailed explanation of any specific step? Let me know!
Here are 5 related questions to expand your understanding:

1. How do you compute the composite of non-polynomial functions?
2. What happens to the domain if $f(x)$ or $g(x)$ contains a square root?
3. How do you graph the resulting composite functions?
4. How do you find the range of these composite functions?
5. Can composite functions result in undefined values for certain $x$?

Tip: Always ensure to check the domain of both functions before combining them, especially when dealing with fractions or square roots!