g•f)(-1).          f(x)= ×²-1.         G(x)= 1/x

To find the composite function \((g \circ f)(-1)\), we need to evaluate \(g(f(-1))\).

Given:
1. \(f(x) = x^2 - 1\)
2. \(g(x) = \frac{1}{x}\)

### Step-by-Step Solution

1. **Find \(f(-1)\):**
   \[
   f(-1) = (-1)^2 - 1 = 1 - 1 = 0
   \]

2. **Substitute \(f(-1)\) into \(g(x)\):**
   Since \(f(-1) = 0\), we need to find \(g(0)\).

   \[
   g(0) = \frac{1}{0}
   \]

   The expression \(\frac{1}{0}\) is undefined.

### Conclusion
\((g \circ f)(-1)\) is undefined because \(g(0)\) involves division by zero, which is not allowed in mathematics.

Would you like more details, or do you have any questions?

Here are 5 related questions you might find interesting:
1. What is the domain of the composite function \(g \circ f(x)\)?
2. How do you find the range of a composite function?
3. How would you evaluate \((f \circ g)(x)\) for different values of \(x\)?
4. What happens if \(f(x)\) or \(g(x)\) have a restricted domain?
5. How does the composition of two functions affect their respective domains?

**Tip:** When working with composite functions, always check the domains of the individual functions to ensure that the composition is defined!

Learn how to evaluate the composite function g ∘ f(-1) using function evaluation principles and understand why it is undefined due to division by zero. Suitable for high school students in Grades 10-12.

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