let f(x)=$$ \sqrt{x^2-1} $$ And g(x)=$$ \frac{1}{\sqrt{4-x^2}} $$ Find (fg)(x)

Given the functions $ f(x) = \sqrt{x^2 - 1} $ and $ g(x) = \frac{1}{\sqrt{4 - x^2}} $, we need to find the product function $ (fg)(x) $.

The product of two functions $ f(x) $ and $ g(x) $ is defined as:
\[
(fg)(x) = f(x) \cdot g(x)
\]

Substituting the given functions:
\[
(fg)(x) = \sqrt{x^2 - 1} \cdot \frac{1}{\sqrt{4 - x^2}}
\]

This simplifies to:
\[
(fg)(x) = \frac{\sqrt{x^2 - 1}}{\sqrt{4 - x^2}}
\]

Therefore, the function $ (fg)(x) $ is:
\[
(fg)(x) = \frac{\sqrt{x^2 - 1}}{\sqrt{4 - x^2}}
\]

### Domain Considerations:
- For $ f(x) = \sqrt{x^2 - 1} $ to be defined, $ x^2 - 1 \geq 0 $, which means $ x \leq -1 $ or $ x \geq 1 $.
- For $ g(x) = \frac{1}{\sqrt{4 - x^2}} $ to be defined, $ 4 - x^2 > 0 $, which means $ -2 < x < 2 $.

The domain of $ (fg)(x) $ is the intersection of the domains of $ f(x) $ and $ g(x) $:
\[
\text{Domain of } (fg)(x) = [-2, -1] \cup [1, 2]
\]

Would you like a detailed explanation of any part of this process?

### Related Questions:
1. What is the domain of $ f(x) = \sqrt{x^2 - 1} $?
2. How do you find the domain of $ g(x) = \frac{1}{\sqrt{4 - x^2}} $?
3. Can you graph the functions $ f(x) $ and $ g(x) $?
4. How would you find the zeros of the function $ (fg)(x) $?
5. How does the domain restriction affect the behavior of $ (fg)(x) $?

### Tip:
When combining functions, always pay close attention to the domains to avoid including values that make the functions undefined.

Learn how to find the product of functions f(x) = sqrt(x^2 - 1) and g(x) = 1/sqrt(4 - x^2) with a step-by-step solution. Explore the domain considerations and mathematical operations involved in combining these functions. Suitable for students in Grades 11-12.

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