To find $(f \circ g)(x)$ and $(g \circ f)(x)$, we are working with function composition. Let’s solve each step by step.

We are given:

• $f(x) = 2x^2 - 6$
• $g(x) = 3x$

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Part 1: $(f \circ g)(x)$

By definition, $(f \circ g)(x) = f(g(x))$.
Here’s how we proceed:

1. Start with $g(x)$: $g(x) = 3x$

2. Substitute $g(x)$ into $f(x)$, which is $f(u) = 2u^2 - 6$. Replace $u$ with $g(x) = 3x$: $f(g(x)) = 2(3x)^2 - 6$

3. Simplify: $f(g(x)) = 2 \cdot (9x^2) - 6$ $f(g(x)) = 18x^2 - 6$

Thus: $(f \circ g)(x) = 18x^2 - 6$

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Part 2: $(g \circ f)(x)$

By definition, $(g \circ f)(x) = g(f(x))$.
Here’s how we proceed:

1. Start with $f(x)$: $f(x) = 2x^2 - 6$

2. Substitute $f(x)$ into $g(x)$, which is $g(u) = 3u$. Replace $u$ with $f(x) = 2x^2 - 6$: $g(f(x)) = 3(2x^2 - 6)$

3. Simplify: $g(f(x)) = 3 \cdot 2x^2 - 3 \cdot 6$ $g(f(x)) = 6x^2 - 18$

Thus: $(g \circ f)(x) = 6x^2 - 18$

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Final Answer:

1. $(f \circ g)(x) = 18x^2 - 6$
2. $(g \circ f)(x) = 6x^2 - 18$

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Do you want a step-by-step breakdown of how I substituted and simplified? Let me know!

Related Questions:

1. What is the difference between $(f \circ g)(x)$ and $(g \circ f)(x)$?
2. How do you find the domain of $(f \circ g)(x)$?
3. If $h(x) = x^2 + 2$, what is $(h \circ g)(x)$?
4. How does composition of functions relate to transformations of graphs?
5. Can $(f \circ g)(x)$ ever equal $(g \circ f)(x)$ for certain functions?

Tip: Always substitute one function completely into the other before simplifying!