To find $(g \circ f)(x)$ and $(f \circ g)(x)$, we need to compute the compositions of the functions $f(x) = x + 5$ and $g(x) = \frac{1}{x}$.

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

$(g \circ f)(x) = g(f(x))$ This means we substitute $f(x)$ into $g(x)$.

1. Start with $f(x) = x + 5$.
2. Substitute $x + 5$ into $g(x) = \frac{1}{x}$: $g(f(x)) = g(x + 5) = \frac{1}{x + 5}$

Therefore, $(g \circ f)(x) = \frac{1}{x + 5}$

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

$(f \circ g)(x) = f(g(x))$ This means we substitute $g(x)$ into $f(x)$.

1. Start with $g(x) = \frac{1}{x}$.
2. Substitute $\frac{1}{x}$ into $f(x) = x + 5$: $f(g(x)) = f\left(\frac{1}{x}\right) = \frac{1}{x} + 5$

Therefore, $(f \circ g)(x) = \frac{1}{x} + 5$

Summary:

$(g \circ f)(x) = \frac{1}{x + 5}$

$(f \circ g)(x) = \frac{1}{x} + 5$

Would you like more details or have any questions?

Here are 5 related questions:

1. What are the domains of $(g \circ f)(x)$ and $(f \circ g)(x)$?
2. How do you find the inverse of a composition of two functions?
3. What is the range of $(g \circ f)(x)$?
4. Can you find $(f \circ f)(x)$ and $(g \circ g)(x)$?
5. How do you verify if two functions are inverses of each other?

Tip: When working with function compositions, always check the domains of the individual functions and the composed function to ensure that all values are valid.