if f(x) = √x-5 and g(x)=√-x+5 what is the domain of (f*g)(x)

To find the domain of the function \((f \cdot g)(x)\), we need to consider the domain restrictions of both functions individually, then combine them. Here's the step-by-step process:

### 1. Domain of \( f(x) = \sqrt{x - 5} \)
- The square root function is only defined when the expression inside the root is non-negative.
- So, \( x - 5 \geq 0 \).
- Solving this inequality: 
  \[
  x \geq 5
  \]
- Therefore, the domain of \( f(x) \) is \( [5, \infty) \).

### 2. Domain of \( g(x) = \sqrt{-x + 5} \)
- Again, the square root function is only defined when the expression inside the root is non-negative.
- So, \( -x + 5 \geq 0 \), or equivalently \( x \leq 5 \).
- Therefore, the domain of \( g(x) \) is \( (-\infty, 5] \).

### 3. Domain of \( (f \cdot g)(x) \)
- The domain of \( (f \cdot g)(x) \) is the intersection of the domains of \( f(x) \) and \( g(x) \), because both functions need to be defined at the same time for their product to exist.
- The domain of \( f(x) \) is \( [5, \infty) \) and the domain of \( g(x) \) is \( (-\infty, 5] \).
- The intersection of these two domains is the single point \( x = 5 \).

### Final Answer:
The domain of \( (f \cdot g)(x) \) is \( \{5\} \).

Would you like further details or have any other questions?

Here are 5 related questions to deepen your understanding:
1. What is the range of \( (f \cdot g)(x) \) given the domain?
2. What happens to the domain if the functions are added instead of multiplied?
3. Can \( f(x) \) and \( g(x) \) ever be negative for any real number?
4. What is the value of \( (f \cdot g)(x) \) at \( x = 5 \)?
5. How would the domain change if \( f(x) = \sqrt{x + 5} \)?

**Tip**: Always check the domain restrictions imposed by the function types (e.g., square roots require non-negative values inside).

Discover how to determine the domain of (f * g)(x) where f(x) = √(x - 5) and g(x) = √(-x + 5). Learn the step-by-step solution by analyzing the domain restrictions of each function and finding their intersection. Ideal for students in Grades 9-11.

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