To determine $(f \circ g)(x)$ and $(g \circ f)(x)$, let’s go step by step.

We are given:

• $f(x) = -3x^2 + 1$
• $g(x) = 3x$

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

The composition $(f \circ g)(x)$ means $f(g(x))$. To compute this, we substitute $g(x)$ into $f(x)$.

• Start with $g(x) = 3x$.
• Now, replace $x$ in $f(x)$ with $g(x)$.

$f(g(x)) = f(3x) = -3(3x)^2 + 1$

Simplify the expression:

$f(g(x)) = -3(9x^2) + 1$

$f(g(x)) = -27x^2 + 1$

Thus:

$(f \circ g)(x) = -27x^2 + 1$

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

The composition $(g \circ f)(x)$ means $g(f(x))$. To compute this, we substitute $f(x)$ into $g(x)$.

• Start with $f(x) = -3x^2 + 1$.
• Now, replace $x$ in $g(x)$ with $f(x)$.

$g(f(x)) = g(-3x^2 + 1) = 3(-3x^2 + 1)$

Simplify the expression:

$g(f(x)) = -9x^2 + 3$

Thus:

$(g \circ f)(x) = -9x^2 + 3$

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Final Results:

1. $(f \circ g)(x) = -27x^2 + 1$
2. $(g \circ f)(x) = -9x^2 + 3$

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Do you need further explanation or examples? Let me know!

Here are some related questions to expand your understanding:

1. How do you evaluate $(f \circ g)(2)$ and $(g \circ f)(2)$ for this example?
2. What is the domain of $(f \circ g)(x)$ and $(g \circ f)(x)$?
3. If $f(x) = -3x^2 + 1$ and $g(x) = 3x + 4$, how would the compositions change?
4. Can you graph $f(x)$, $g(x)$, and their compositions?
5. What is the difference between $(f \circ g)(x)$ and $(g \circ f)(x)$?

Tip:

The order of composition matters! $(f \circ g)(x) \neq (g \circ f)(x)$ in most cases. Always substitute carefully!