Given functions:

• $f(x) = x - 3$
• $g(x) = 5x^2 - 2$

We want to find:

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

Let's solve each part step-by-step.

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

The notation $(f \circ g)(x)$ represents the composite function $f(g(x))$, where we substitute $g(x)$ into $f(x)$.

1. Substitute $g(x) = 5x^2 - 2$ into $f(x)$: $f(g(x)) = f(5x^2 - 2)$
2. Since $f(x) = x - 3$, we replace $x$ in $f(x)$ with $5x^2 - 2$: $f(5x^2 - 2) = (5x^2 - 2) - 3$
3. Simplify: $f(g(x)) = 5x^2 - 5$

Thus, $(f \circ g)(x) = 5x^2 - 5$.

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

The notation $(g \circ f)(x)$ represents the composite function $g(f(x))$, where we substitute $f(x)$ into $g(x)$.

1. Substitute $f(x) = x - 3$ into $g(x)$: $g(f(x)) = g(x - 3)$
2. Since $g(x) = 5x^2 - 2$, we replace $x$ in $g(x)$ with $x - 3$: $g(x - 3) = 5(x - 3)^2 - 2$
3. Expand $(x - 3)^2$: $(x - 3)^2 = x^2 - 6x + 9$
4. Substitute back: $g(x - 3) = 5(x^2 - 6x + 9) - 2$
5. Distribute and simplify: $g(x - 3) = 5x^2 - 30x + 45 - 2$ $g(x - 3) = 5x^2 - 30x + 43$

Thus, $(g \circ f)(x) = 5x^2 - 30x + 43$.

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

Now, we substitute $x = 1$ into $(f \circ g)(x)$.

1. First, find $g(1)$: $g(1) = 5(1)^2 - 2 = 5 - 2 = 3$
2. Then, substitute $g(1) = 3$ into $f(x)$: $f(g(1)) = f(3)$
3. Calculate $f(3)$: $f(3) = 3 - 3 = 0$

Thus, $(f \circ g)(1) = 0$.

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

Finally, substitute $x = 1$ into $(g \circ f)(x)$.

1. First, find $f(1)$: $f(1) = 1 - 3 = -2$
2. Then, substitute $f(1) = -2$ into $g(x)$: $g(f(1)) = g(-2)$
3. Calculate $g(-2)$: $g(-2) = 5(-2)^2 - 2 = 5 \cdot 4 - 2 = 20 - 2 = 18$

Thus, $(g \circ f)(1) = 18$.

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Summary of Results

• (a) $(f \circ g)(x) = 5x^2 - 5$
• (b) $(g \circ f)(x) = 5x^2 - 30x + 43$
• (c) $(f \circ g)(1) = 0$
• (d) $(g \circ f)(1) = 18$

Would you like any details on these steps or have questions?

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Further Related Questions

1. What is the domain of $(f \circ g)(x)$?
2. What is the range of $(f \circ g)(x)$?
3. How would the composition change if $f(x)$ were a different linear function, like $f(x) = 2x + 1$?
4. Can composite functions be commutative for all functions?
5. How do you find the inverse of $f(g(x))$?

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Tip:

When forming composite functions, always work step-by-step to substitute and simplify, ensuring each inner function is fully evaluated before moving to the next.