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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!

Find the composition of the functions F(x) = x^2 - 1 and g(x) = 1/x, and determine (F∘g)(x).

Learn how to find the composition of functions F(x) = x^2 - 1 and g(x) = 1/x. This algebra problem breaks down the steps to calculate F(g(x)) and explores function compositions. Suitable for Grades 10-12.

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