We are tasked with finding $(f \circ g)(x)$ and $(g \circ f)(x)$, where $f(x) = 3x^2 + 5$ and $g(x) = -3x$.

1. Finding $(f \circ g)(x)$:

The composition $(f \circ g)(x)$ means $f(g(x))$. Substituting $g(x) = -3x$ into $f(x)$:

$f(g(x)) = f(-3x) = 3(-3x)^2 + 5$

Simplify:

$(-3x)^2 = 9x^2 \quad \text{so} \quad 3(-3x)^2 = 3 \cdot 9x^2 = 27x^2$

$f(-3x) = 27x^2 + 5$

Thus:

$(f \circ g)(x) = 27x^2 + 5$

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2. Finding $(g \circ f)(x)$:

The composition $(g \circ f)(x)$ means $g(f(x))$. Substituting $f(x) = 3x^2 + 5$ into $g(x)$:

$g(f(x)) = g(3x^2 + 5) = -3(3x^2 + 5)$

Simplify:

$-3(3x^2 + 5) = -9x^2 - 15$

Thus:

$(g \circ f)(x) = -9x^2 - 15$

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Final Answer:

$(f \circ g)(x) = 27x^2 + 5$ $(g \circ f)(x) = -9x^2 - 15$

Would you like a detailed breakdown of any steps or additional examples?

Related Questions:

1. How do you verify the results of function compositions?
2. What are the domain restrictions of $(f \circ g)(x)$ and $(g \circ f)(x)$?
3. How does the order of composition affect the result?
4. Can you apply these compositions to specific values of $x$?
5. What happens when both $f(x)$ and $g(x)$ are linear functions?

Tip:

Always simplify expressions inside parentheses before applying the outer function in compositions.