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The problem asks to find the compositions of two functions \( (f \circ g)(x) \) and \( (g \circ f)(x) \), as well as the domain of each. The functions given are:

\[
f(x) = x + 10, \quad g(x) = x - 10.
\]

### Step 1: Find \( (f \circ g)(x) \)

The composition \( (f \circ g)(x) \) means we apply \( g(x) \) first and then apply \( f \) to the result. Mathematically, this is:

\[
(f \circ g)(x) = f(g(x)).
\]

Substitute \( g(x) = x - 10 \) into \( f(x) \):

\[
f(g(x)) = f(x - 10) = (x - 10) + 10 = x.
\]

So, \( (f \circ g)(x) = x \).

### Step 2: Find \( (g \circ f)(x) \)

The composition \( (g \circ f)(x) \) means we apply \( f(x) \) first and then apply \( g \) to the result. Mathematically, this is:

\[
(g \circ f)(x) = g(f(x)).
\]

Substitute \( f(x) = x + 10 \) into \( g(x) \):

\[
g(f(x)) = g(x + 10) = (x + 10) - 10 = x.
\]

So, \( (g \circ f)(x) = x \).

### Step 3: Determine the Domains

For both \( (f \circ g)(x) = x \) and \( (g \circ f)(x) = x \), the domains of \( f(x) \) and \( g(x) \) are all real numbers (\( \mathbb{R} \)). Since there are no restrictions in either composition that would limit the domain, the domain of both \( (f \circ g)(x) \) and \( (g \circ f)(x) \) is also all real numbers.

### Final Answer:

- \( (f \circ g)(x) = x \), Domain: \( \mathbb{R} \)
- \( (g \circ f)(x) = x \), Domain: \( \mathbb{R} \)

Would you like more details on any of these steps or have any questions? Here are some related questions you might consider:

1. What is the general method to find the composition of two functions?
2. How do you find the domain of a function composition?
3. What happens to the domain when one of the functions in the composition has a restricted domain?
4. Can compositions of non-linear functions also result in linear functions?
5. How would the compositions change if the functions were \( f(x) = x^2 + 10 \) and \( g(x) = \sqrt{x - 10} \)?

**Tip:** Always remember to check the domain when composing functions, especially when dealing with square roots, logarithms, or rational expressions that have restrictions.

Find (f ∘ g)(x) and (g ∘ f)(x) and the domain of each. f(x) = x + 10, g(x) = x - 10

Learn how to find the compositions of two functions f(x) = x + 10 and g(x) = x - 10. This guide covers key algebraic concepts, including function composition and determining domains for each function.

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