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

To find \((g \circ f)(-1)\), we need to evaluate \(g(f(-1))\). This means we first find the value of \(f(-1)\) and then plug that result into \(g(x)\).

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

### Step-by-Step Solution

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

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

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

3. **Determine if \(g(0)\) is defined:**
   Division by zero is undefined in mathematics. Therefore:

### Conclusion
\((g \circ f)(-1)\) is **undefined** because it involves dividing by zero when evaluating \(g(0)\).

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 function \(g \circ f(x)\)?
2. How would you find the composition \((f \circ g)(x)\) for different values of \(x\)?
3. What does it mean for a function to be undefined?
4. How can you determine if a composite function is continuous?
5. How does the composition of functions affect the overall function's behavior?

**Tip:** Always check for values that cause division by zero when dealing with rational functions to avoid undefined expressions!

Learn how to evaluate the composite function (g ∘ f)(-1) using the functions f(x) = x² - 1 and g(x) = 1/x. Understand why the result is undefined due to division by zero when evaluating g(0). Suitable for high school students studying function composition and rational functions.

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