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It looks like you've uploaded an image containing two homework problems related to function composition. Let me go over both problems and their solutions in detail:

### Problem 1:

Let \( f \), \( g \), and \( h \) be functions from \( \mathbb{Z} \) (the set of integers) to \( \mathbb{Z} \), defined as:
- \( f(x) = x + 5 \)
- \( g(x) = x - 2 \)
- \( h(x) = x^3 \)

We need to find the following compositions:
- \( f \circ g \)
- \( g \circ f \)
- \( h \circ g \)
- \( g \circ h \)
- \( f \circ h \)
- \( h \circ f \)

**Solution:**
1. \( f \circ g = f(g(x)) = f(x - 2) = (x - 2) + 5 = x + 3 \)
2. \( g \circ f = g(f(x)) = g(x + 5) = (x + 5) - 2 = x + 3 \)
3. \( h \circ g = h(g(x)) = h(x - 2) = (x - 2)^3 \)
4. \( g \circ h = g(h(x)) = g(x^3) = x^3 - 2 \)
5. \( f \circ h = f(h(x)) = f(x^3) = x^3 + 5 \)
6. \( h \circ f = h(f(x)) = h(x + 5) = (x + 5)^3 \)

### Problem 2:

Let \( f \) and \( g \) be functions from \( \mathbb{R} \) (the set of real numbers) to \( \mathbb{R} \), defined as:
- \( f(x) = x - 1 \)
- \( g(x) = x + 1 \)

We need to find the following compositions:
- \( f \circ g \)
- \( g \circ f \)
- \( f \circ f \)
- \( g \circ g \)

**Solution:**
1. \( f \circ g = f(g(x)) = f(x + 1) = (x + 1) - 1 = x \)
2. \( g \circ f = g(f(x)) = g(x - 1) = (x - 1) + 1 = x \)
3. \( f \circ f = f(f(x)) = f(x - 1) = (x - 1) - 1 = x - 2 \)
4. \( g \circ g = g(g(x)) = g(x + 1) = (x + 1) + 1 = x + 2 \)

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### Do you have any specific questions, or would you like further details on any part of these solutions?

Here are 5 related questions to expand on the topic:
1. What would the composition \( h \circ h \) be for the first set of functions?
2. Can we generalize the results of \( f \circ f \) and \( g \circ g \) for arbitrary functions \( f(x) = x + c \) and \( g(x) = x - c \)?
3. How would the results change if \( f(x) \) and \( g(x) \) were non-linear functions (e.g., \( f(x) = x^2 \))?
4. What are the conditions under which \( f \circ g = g \circ f \)?
5. Can you explain the inverse function concept in the context of these function compositions?

**Tip:** When composing functions, always evaluate the inner function first and then apply the outer function. This order is critical in obtaining the correct result.

1. Let f, g, h be functions from Z to Z defined as f(x) = x + 5, g(x) = x - 2, h(x) = x^3 for all x ∈ Z. Find f ∘ g, g ∘ f, h ∘ g, g ∘ h, f ∘ h, h ∘ f. 2. Let f: R → R defined as f(x) = x - 1 for all x ∈ R and g: R → R defined as g(x) = x + 1 for all x ∈ R. Find f ∘ g, g ∘ f, f ∘ f, and g ∘ g.

Learn how to solve function composition problems involving functions defined over integers and real numbers. This detailed guide covers compositions like f ∘ g, g ∘ f, h ∘ g, g ∘ h, f ∘ h, and h ∘ f with solutions. Suitable for high school students in grades 9-12.

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