The problem asks for the composition of two functions, $F(x) = x^2 - 1$ and $g(x) = \frac{1}{x}$, where we are tasked with finding $(F \circ g)(x)$, which is the composition of $F$ and $g$, also written as $F(g(x))$.

Let's break it down:

1. First, identify $g(x)$: $g(x) = \frac{1}{x}$

2. Now substitute $g(x)$ into $F(x)$: $F(g(x)) = F\left( \frac{1}{x} \right)$ Since $F(x) = x^2 - 1$, we replace $x$ with $\frac{1}{x}$: $F\left( \frac{1}{x} \right) = \left( \frac{1}{x} \right)^2 - 1 = \frac{1}{x^2} - 1$

Thus, the composition $(F \circ g)(x) = F(g(x))$ is: $(F \circ g)(x) = \frac{1}{x^2} - 1$

Would you like more details, or do you have any further questions? Here are some related questions you might consider:

1. How do you find the domain of a composite function?
2. What happens if you compose $g \circ F(x)$ instead of $F \circ g(x)$?
3. How do you handle compositions with trigonometric functions?
4. What are common mistakes when composing functions?
5. How can we visually interpret function compositions?

Tip: When composing functions, always substitute carefully, paying close attention to where the inner function replaces the variable in the outer function!