please explain how to find domain of compositions of functions simply and provide example

To find the domain of a composition of functions \( (f \circ g)(x) = f(g(x)) \), follow these simple steps:

### Steps to Find the Domain of a Composition of Functions:

1. **Find the domain of the inner function \( g(x) \):** 
   - The domain of \( g(x) \) consists of all \( x \)-values for which \( g(x) \) is defined.

2. **Find the domain of the outer function \( f(x) \):**
   - This involves ensuring that the values output by \( g(x) \) (i.e., the inputs to \( f \)) must be valid for \( f(x) \). So, find where \( f \) is defined.

3. **Combine the conditions:** 
   - The domain of \( f(g(x)) \) is all \( x \)-values for which \( g(x) \) is defined **and** for which \( f(g(x)) \) is defined.

### Example:
Let \( f(x) = \sqrt{x} \) and \( g(x) = x^2 - 1 \). Find the domain of \( (f \circ g)(x) = f(g(x)) = \sqrt{x^2 - 1} \).

1. **Domain of the inner function \( g(x) = x^2 - 1 \):**
   - \( g(x) \) is a polynomial, so it is defined for all real numbers, i.e., \( (-\infty, \infty) \).

2. **Domain of the outer function \( f(x) = \sqrt{x} \):**
   - The square root function \( f(x) = \sqrt{x} \) is defined only when \( x \geq 0 \).

3. **Combine the conditions:**
   - We need \( g(x) = x^2 - 1 \geq 0 \), because \( f(g(x)) = \sqrt{x^2 - 1} \).
   - Solve \( x^2 - 1 \geq 0 \):
     \[
     x^2 \geq 1 \quad \Rightarrow \quad x \geq 1 \text{ or } x \leq -1
     \]
   - So, the domain of \( (f \circ g)(x) \) is \( (-\infty, -1] \cup [1, \infty) \).

### Conclusion:
The domain of the composition \( (f \circ g)(x) \) is \( (-\infty, -1] \cup [1, \infty) \).

Would you like further details or have any questions? Here are some related questions to expand this understanding:

1. What happens if the outer function has a more restrictive domain?
2. How do you handle compositions with rational functions?
3. Can you find the domain for \( (f \circ g)(x) \) when \( g(x) \) is a trigonometric function?
4. What is the domain of a composition where the outer function is a logarithmic function?
5. How does the range of the inner function affect the composition's domain?

**Tip:** Always pay close attention to whether the inner function produces values that are compatible with the domain of the outer function.

Learn how to find the domain of composed functions using a simple step-by-step guide. This example involves the composition of functions f(x) = √x and g(x) = x² - 1, explaining how to combine domain restrictions. Suitable for high school students.

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