To solve this problem, we compute the compositions of the given functions $f(x) = x+2$, $g(x) = x-2$, and $h(x) = 3x$ as specified. Let’s calculate each step by step:

1. Compute $(g \circ f)(x)$:

This means $g(f(x))$, so substitute $f(x) = x+2$ into $g(x) = x-2$:

$g(f(x)) = g(x+2) = (x+2) - 2 = x.$

Thus, $(g \circ f)(x) = x$.

2. Compute $(f \circ g)(x)$:

This means $f(g(x))$, so substitute $g(x) = x-2$ into $f(x) = x+2$:

$f(g(x)) = f(x-2) = (x-2) + 2 = x.$

Thus, $(f \circ g)(x) = x$.

3. Compute $(f \circ f)(x)$:

This means $f(f(x))$, so substitute $f(x) = x+2$ into itself:

$f(f(x)) = f(x+2) = (x+2) + 2 = x + 4.$

Thus, $(f \circ f)(x) = x + 4$.

4. Compute $(g \circ g)(x)$:

This means $g(g(x))$, so substitute $g(x) = x-2$ into itself:

$g(g(x)) = g(x-2) = (x-2) - 2 = x - 4.$

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

Final Results:

1. $(g \circ f)(x) = x$
2. $(f \circ g)(x) = x$
3. $(f \circ f)(x) = x + 4$
4. $(g \circ g)(x) = x - 4$

Would you like a detailed explanation of any of these calculations?

5 Related Questions:

1. What are the properties of composition functions such as $f \circ g$ and $g \circ f$?
2. How does the identity function relate to the compositions $(g \circ f)(x) = x$ and $(f \circ g)(x) = x$?
3. Can we compute higher-order compositions, such as $(f \circ f \circ f)(x)$?
4. What happens if we compose $h(x)$ with $f(x)$ or $g(x)$?
5. How do these compositions affect the domain and range of the resulting functions?

Tip:

Always evaluate the inner function first when working with compositions. This ensures accuracy and clarity in computations.