https://r2.erweima.ai/mathbot/image/1727146502244nvf5bqpw.jpg

The problem in the image asks to evaluate composite functions based on the given functions \( f(x) \) and \( g(x) \):

- \( f(x) = \sqrt{x} \)
- \( g(x) = x^2 - 1 \)

We are tasked with finding the following:

1. \( f(g(x)) \)
2. \( g(f(x)) \)

### Step-by-Step Solutions:

1. **Evaluate \( f(g(x)) \):**
   - First, compute \( g(x) \):  
     \( g(x) = x^2 - 1 \)
   - Now substitute \( g(x) \) into \( f(x) \):  
     \( f(g(x)) = f(x^2 - 1) \)
   - Since \( f(x) = \sqrt{x} \), we get:  
     \( f(g(x)) = \sqrt{x^2 - 1} \)

2. **Evaluate \( g(f(x)) \):**
   - First, compute \( f(x) \):  
     \( f(x) = \sqrt{x} \)
   - Now substitute \( f(x) \) into \( g(x) \):  
     \( g(f(x)) = g(\sqrt{x}) \)
   - Since \( g(x) = x^2 - 1 \), substitute \( \sqrt{x} \) for \( x \):  
     \( g(f(x)) = (\sqrt{x})^2 - 1 = x - 1 \)

### Final Answers:
1. \( f(g(x)) = \sqrt{x^2 - 1} \)
2. \( g(f(x)) = x - 1 \)

Would you like more details on these calculations or have any questions?

Here are some related questions:
1. What are the general rules for evaluating composite functions?
2. How do the domains of \( f(x) \) and \( g(x) \) affect the domain of \( f(g(x)) \)?
3. What happens if the input to \( f(x) \) or \( g(x) \) is negative?
4. How do composite functions relate to transformations of graphs?
5. Can you generalize the composition of two arbitrary functions?

**Tip**: When dealing with composite functions, always solve the inner function first and then apply the outer function.

Evaluar funciones compuestas: dadas f(x) = √x y g(x) = x² - 1, evalúe f(g(x)) y g(f(x)).

Learn how to evaluate composite functions, including f(g(x)) and g(f(x)), where f(x) = √x and g(x) = x² - 1. This example covers the key concepts of composite functions and how to handle square roots and quadratic functions.

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