We are tasked with finding two compositions of functions, one for $(f \circ g)(x)$ and another for $(g \circ f)(4)$. Let's break this down step by step:

Given functions:

• $f(x) = -2x^2 - 3x + 8$
• $g(x) = 2x - 5$

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

This represents $f(g(x))$, meaning we will substitute $g(x)$ into the function $f(x)$.

Step 1: Substitute $g(x) = 2x - 5$ into $f(x)$.

The function $f(x) = -2x^2 - 3x + 8$ becomes: $f(g(x)) = f(2x - 5)$

Step 2: Replace $x$ in $f(x)$ with $2x - 5$:

$f(2x - 5) = -2(2x - 5)^2 - 3(2x - 5) + 8$

Step 3: Simplify each term:

• First, simplify $(2x - 5)^2$: $(2x - 5)^2 = (2x)^2 - 2(2x)(5) + (5)^2 = 4x^2 - 20x + 25$
• Now substitute this result back into the equation: $f(2x - 5) = -2(4x^2 - 20x + 25) - 3(2x - 5) + 8$

Step 4: Distribute:

$f(2x - 5) = -2(4x^2) + 2(20x) - 2(25) - 3(2x) + 3(5) + 8$ $= -8x^2 + 40x - 50 - 6x + 15 + 8$

Step 5: Combine like terms:

$f(2x - 5) = -8x^2 + 34x - 27$

Thus, the composition $(f \circ g)(x) = -8x^2 + 34x - 27$.

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

This represents $g(f(4))$, meaning we first evaluate $f(4)$, then substitute that result into $g(x)$.

Step 1: Find $f(4)$.

Substitute $x = 4$ into $f(x)$: $f(4) = -2(4)^2 - 3(4) + 8$ $= -2(16) - 12 + 8 = -32 - 12 + 8 = -36$

Step 2: Find $g(f(4)) = g(-36)$.

Now substitute $-36$ into $g(x)$: $g(-36) = 2(-36) - 5 = -72 - 5 = -77$

Thus, $(g \circ f)(4) = -77$.

Final Answers:

• $(f \circ g)(x) = -8x^2 + 34x - 27$
• $(g \circ f)(4) = -77$

Would you like further details or explanations?

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Here are five related questions to expand on this topic:

1. How do you find the domain of the composition of two functions?
2. What is the difference between $f(g(x))$ and $g(f(x))$?
3. Can compositions of functions be commutative (i.e., $f(g(x)) = g(f(x))$)?
4. How does function composition relate to real-world applications, such as in physics or economics?
5. What happens to the graph of a function when it's composed with another function?

Tip: Always simplify inner functions first when working with compositions to avoid errors!