find the f x f in f(x)= square root of x, g(x)= x^2-1

To find \( f(g(x)) \) where \( f(x) = \sqrt{x} \) and \( g(x) = x^2 - 1 \), you need to substitute the function \( g(x) \) into the function \( f(x) \).

### Step-by-Step Solution:

1. **Identify the functions**:
   - \( f(x) = \sqrt{x} \)
   - \( g(x) = x^2 - 1 \)

2. **Substitute \( g(x) \) into \( f(x) \)**:
   - \( f(g(x)) = f(x^2 - 1) \)

3. **Apply the function \( f(x) \)**:
   - \( f(g(x)) = \sqrt{x^2 - 1} \)

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

Would you like more details on this process or have any further questions?

### Related Questions:
1. What is \( g(f(x)) \) if \( g(x) = x^2 - 1 \) and \( f(x) = \sqrt{x} \)?
2. How do you find the domain of \( f(g(x)) \) when \( f(x) = \sqrt{x} \) and \( g(x) = x^2 - 1 \)?
3. Can you calculate the derivative of \( f(g(x)) \) when \( f(x) = \sqrt{x} \) and \( g(x) = x^2 - 1 \)?
4. How would you find the inverse of \( f(x) = \sqrt{x} \)?
5. How does the graph of \( f(g(x)) = \sqrt{x^2 - 1} \) compare to the graph of \( g(x) = x^2 - 1 \)?

### Tip:
When composing functions, always make sure the output of the inner function is within the domain of the outer function to avoid undefined values.

Explore how to compute \( f(g(x)) \) where \( f(x) = \sqrt{x} \) and \( g(x) = x^2 - 1 \). Step-by-step solution provided with detailed explanations. Suitable for advanced high school students studying function composition and quadratic functions.

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