1. Given 𝑓(𝑥) = −2𝑥 2 − 3𝑥 + 8 𝑔(𝑥) = 2𝑥 − 5
Find the indicated composition. Show all work.
a. (𝑓 ∘ 𝑔)(𝑥) b. (𝑔 ∘ 𝑓)(4)

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!

1. Given 𝑓(𝑥) = −2𝑥^2 − 3𝑥 + 8 and 𝑔(𝑥) = 2𝑥 − 5, find the indicated composition. Show all work. a. (𝑓 ∘ 𝑔)(𝑥) b. (𝑔 ∘ 𝑓)(4)

Learn how to solve the function composition problem involving f(x) = −2x^2 − 3x + 8 and g(x) = 2x − 5. This detailed solution covers how to find (f ∘ g)(x) and (g ∘ f)(4) using algebraic techniques such as function substitution and quadratic expansion. Suitable for students in Grades 10-12.

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