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. $g \circ g = g(g(x)) = g(x + 1) = (x + 1) + 1 = x + 2$

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.