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The question involves the function \( f(x) = \frac{1}{5x^2 + 1} \) for \( x \geq 0 \), and it asks for the graph of its inverse function, \( f^{-1}(x) \), using symmetry with respect to the line \( y = x \).

### Analysis

1. **Symmetry and Graph Selection**:  
   To find the inverse function graphically, we need to reflect the graph of \( f(x) \) across the line \( y = x \). The correct graph must have this reflection property. 

   Observing the options:
   - **Graph C** correctly shows the function \( f(x) \) (in blue) and its inverse \( f^{-1}(x) \) (in red) as symmetric with respect to the line \( y = x \). The graph passes through the point (1, 1), which is a key characteristic of an inverse relationship.

2. **Domain and Range Identification**:
   - The **domain** of \( f(x) = \frac{1}{5x^2 + 1} \) is \( x \geq 0 \), or \( [0, \infty) \).
   - The **range** of \( f(x) \) is the set of all values the function can take. Since \( f(x) = \frac{1}{5x^2 + 1} \), it will approach 0 as \( x \to \infty \) and will be at its maximum when \( x = 0 \), giving \( f(0) = 1 \). Hence, the range of \( f(x) \) is \( (0, 1] \).

   The inverse function \( f^{-1}(x) \) will swap these:
   - **Domain of \( f^{-1}(x) \)**: The range of \( f(x) \), which is \( (0, 1] \).
   - **Range of \( f^{-1}(x) \)**: The domain of \( f(x) \), which is \( [0, \infty) \).

### Conclusion

The correct graph is **option C**.  
- **Domain of \( f^{-1}(x) \):** \((0, 1]\).
- **Range of \( f^{-1}(x) \):** \([0, \infty)\).

Would you like more details or have any questions?

### Related Questions:

1. How do you find the inverse of a function algebraically?
2. What are the conditions for a function to have an inverse?
3. How can you determine the range of a function without graphing it?
4. Why is the line \( y = x \) used for finding the inverse graphically?
5. How do you differentiate between the graphs of \( f(x) \) and \( f^{-1}(x) \)?

**Tip**: Always remember that the domain of a function becomes the range of its inverse, and vice versa.

The graph of the function f(x) = 1 / (5x^2 + 1), x ≥ 0, is shown to the right. Use symmetry with respect to the line y = x to add the graph of f^{-1}(x) on the same graph. (It is not necessary to find a formula for f^{-1}(x).) Identify the domain and range of f^{-1}(x).

Learn how to use symmetry with respect to the line y = x to graph the inverse function of f(x) = 1 / (5x^2 + 1). Understand how to identify the domain and range of the inverse function. Suitable for students in Grades 10-12.

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