SOLUTION: Suppose the domain of f is (-1,3). Define the function g by g(x) = f(sqrt(x)/sqrt(-x)). What is the domain of g?

Question 1209617: Suppose the domain of f is (-1,3). Define the function g by
g(x) = f(sqrt(x)/sqrt(-x)).
What is the domain of g?
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Here's how to determine the domain of g:
1. **Analyze the input to f:**
The function g(x) is defined as g(x) = f(√(x)/√(-x)). The input to the function f is the expression √(x)/√(-x).
2. **Consider the square roots:**
For √(x) to be defined, x must be greater than or equal to 0 (x ≥ 0).
For √(-x) to be defined, -x must be greater than or equal to 0, which means x must be less than or equal to 0 (x ≤ 0).
3. **Combine the restrictions:**
The expression √(x)/√(-x) requires *both* x ≥ 0 and x ≤ 0 to be true. The only value that satisfies both conditions is x = 0.
4. **Evaluate the expression at x = 0:**
When x = 0, the expression becomes √(0)/√(-0) = 0/0, which is undefined.
5. **Consider the domain of f:**
Even if the expression √(x)/√(-x) were defined, its value would need to fall within the domain of f, which is (-1, 3).
6. **Determine the domain of g:**
Since the expression √(x)/√(-x) is undefined for all x, the function g(x) is also undefined for all x. Therefore, the domain of g is the empty set, often represented as {}.

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