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The problem asks to find the range of \( g \circ f(x) \), where:

- \( f(x) = \frac{1}{x} \)
- \( g(x) = x + 1 \)

### Step-by-Step Solution:

1. **Composition of Functions**:
   We are looking for \( g(f(x)) \). This is the composition of the two functions.

   \[
   g(f(x)) = g\left(\frac{1}{x}\right) = \frac{1}{x} + 1
   \]

2. **Find the Range**:
   Now, we need to find the range of \( \frac{1}{x} + 1 \).

   - The function \( f(x) = \frac{1}{x} \) is defined for all \( x \neq 0 \).
   - As \( x \to \infty \), \( \frac{1}{x} \to 0 \), so \( \frac{1}{x} + 1 \to 1 \).
   - As \( x \to 0^+ \) (from the right), \( \frac{1}{x} \to \infty \), so \( \frac{1}{x} + 1 \to \infty \).
   - As \( x \to 0^- \) (from the left), \( \frac{1}{x} \to -\infty \), so \( \frac{1}{x} + 1 \to -\infty \).

   Thus, \( \frac{1}{x} + 1 \) can take any real value except 1 (since 1 is approached but never reached). Therefore, the range is:

   \[
   (-\infty, 1) \cup (1, \infty) \quad \text{or} \quad y \neq 1
   \]

### Answer:
The correct option is \( y \neq 1 \).

Let me know if you would like more details, and here are some related questions:

1. How do you find the domain of composite functions?
2. What happens when \( f(x) \) is a rational function?
3. Can you explain more about the behavior of limits in composite functions?
4. What is the graphical interpretation of the function \( g(f(x)) \)?
5. How does the range of a function relate to asymptotes?

**Tip**: When working with composite functions, always find the domain of each individual function first and then analyze how the composition affects the final result.

If f(x) = 1/x and g(x) = x + 1, find the range of g∘f(x).

Learn how to find the range of composite functions like g(f(x)) where f(x) = 1/x and g(x) = x + 1. This solution explores function composition, asymptotic behavior, and the range of rational functions. Suitable for high school students.

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