Let's solve the problem step by step:

Given functions:

• $f(x) = x + 5$
• $g(x) = 2x + 3$

We need to find:

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

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

The composition $(f \circ g)(x)$ means $f(g(x))$. Substitute $g(x) = 2x + 3$ into $f(x) = x + 5$: $f(g(x)) = f(2x + 3) = (2x + 3) + 5 = 2x + 8$

So, $(f \circ g)(x) = 2x + 8$.

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

The composition $(g \circ f)(x)$ means $g(f(x))$. Substitute $f(x) = x + 5$ into $g(x) = 2x + 3$: $g(f(x)) = g(x + 5) = 2(x + 5) + 3 = 2x + 10 + 3 = 2x + 13$

So, $(g \circ f)(x) = 2x + 13$.

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Part 3: $(f \circ g)(2)$

From Part 1, $(f \circ g)(x) = 2x + 8$. Substitute $x = 2$: $(f \circ g)(2) = 2(2) + 8 = 4 + 8 = 12$

So, $(f \circ g)(2) = 12$.

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Part 4: $(g \circ f)(2)$

From Part 2, $(g \circ f)(x) = 2x + 13$. Substitute $x = 2$: $(g \circ f)(2) = 2(2) + 13 = 4 + 13 = 17$

So, $(g \circ f)(2) = 17$.

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Final Answers:

a. $(f \circ g)(x) = 2x + 8$
b. $(g \circ f)(x) = 2x + 13$
c. $(f \circ g)(2) = 12$
d. $(g \circ f)(2) = 17$

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Do you want more details or have any questions?
Here are 5 related questions to explore further:

1. What is the domain of $(f \circ g)(x)$ and $(g \circ f)(x)$?
2. How does the composition of functions differ when $f(x)$ and $g(x)$ are swapped?
3. Can we find the inverse of $(f \circ g)(x)$?
4. What happens if $f(x) = x^2$ and $g(x) = \sqrt{x}$?
5. How would $(f \circ g)(x)$ look if $g(x) = x - 3$?

Tip: Always simplify inside parentheses first when working with compositions!