Let's solve the given problem step by step.

We are given two functions:

• $f(x) = 2x - 1$
• $g(x) = 4x + 5$

Part (a) $f(g(x))$

This means we need to substitute $g(x)$ into $f(x)$:

$f(g(x)) = f(4x + 5)$

Substitute $4x + 5$ into $f(x) = 2x - 1$:

$f(4x + 5) = 2(4x + 5) - 1 = 8x + 10 - 1 = 8x + 9$

So, $f(g(x)) = 8x + 9$.

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Part (b) $g(f(x))$

This means we need to substitute $f(x)$ into $g(x)$:

$g(f(x)) = g(2x - 1)$

Substitute $2x - 1$ into $g(x) = 4x + 5$:

$g(2x - 1) = 4(2x - 1) + 5 = 8x - 4 + 5 = 8x + 1$

So, $g(f(x)) = 8x + 1$.

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Part (c) $f(f(x))$

This means we need to substitute $f(x)$ into $f(x)$:

$f(f(x)) = f(2x - 1)$

Substitute $2x - 1$ into $f(x) = 2x - 1$:

$f(2x - 1) = 2(2x - 1) - 1 = 4x - 2 - 1 = 4x - 3$

So, $f(f(x)) = 4x - 3$.

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Final Answers:

• $f(g(x)) = 8x + 9$
• $g(f(x)) = 8x + 1$
• $f(f(x)) = 4x - 3$

Would you like to dive deeper into any of these steps?

Here are five related questions to expand on these concepts:

1. What happens if we switch the order of composition for any two arbitrary functions?
2. Can you find $g(g(x))$ using the same method?
3. How do we prove that function composition is associative?
4. Can you solve $f(g(x)) = 0$ for $x$?
5. What is the graphical interpretation of function composition?

Tip: When composing functions, always substitute the inner function into the outer function carefully by replacing every occurrence of the variable.