f(x)= x^2-9 for x∈R, x<0 g(x)=10-x/2 for x∈R, x>6 find the domain of fg(x)

We are tasked with finding the domain of the composition \( f(g(x)) \), where:

- \( f(x) = x^2 - 9 \), defined for \( x \in \mathbb{R} \) and \( x < 0 \)
- \( g(x) = 10 - \frac{x}{2} \), defined for \( x \in \mathbb{R} \) and \( x > 6 \)

### Step-by-step solution:

1. **Find the range of \( g(x) \):**
   \[
   g(x) = 10 - \frac{x}{2}
   \]
   Since \( g(x) \) is defined for \( x > 6 \), let's evaluate the behavior of \( g(x) \) as \( x \to 6^+ \) and as \( x \to \infty \).

   - When \( x = 6 \), \( g(6) = 10 - \frac{6}{2} = 7 \)
   - As \( x \to \infty \), \( g(x) \to -\infty \)

   Thus, the range of \( g(x) \) is \( (-\infty, 7) \).

2. **Domain of \( f(x) \):**
   The function \( f(x) = x^2 - 9 \) is defined only for \( x < 0 \).

3. **Determine the valid input values for \( g(x) \) that result in \( f(g(x)) \):**
   For \( f(g(x)) \) to be defined, the output of \( g(x) \) must lie in the domain of \( f(x) \), i.e., \( g(x) < 0 \).

   So, we need to solve for \( x \) such that:
   \[
   g(x) = 10 - \frac{x}{2} < 0
   \]
   Solving this inequality:
   \[
   10 - \frac{x}{2} < 0
   \]
   \[
   10 < \frac{x}{2}
   \]
   \[
   x > 20
   \]

   Therefore, \( f(g(x)) \) is defined for \( x > 20 \).

### Conclusion:
The domain of \( f(g(x)) \) is \( x > 20 \).

---

Would you like further clarification or details on any step?

Here are 5 related questions you might find helpful:
1. How do you find the range of a composition of functions?
2. How does the domain of a function affect its composition with another function?
3. What are the different methods to solve inequalities like \( 10 - \frac{x}{2} < 0 \)?
4. Can you graph \( g(x) \) to visualize how its range intersects with the domain of \( f(x) \)?
5. How would the result change if \( f(x) \) had a different domain?

**Tip:** When dealing with compositions, always check the range of the inner function and the domain of the outer function!

f(x) = x^2 - 9 for x ∈ ℝ, x < 0; g(x) = 10 - x/2 for x ∈ ℝ, x > 6. Find the domain of f(g(x)).

This detailed guide covers how to find the domain of the composite function f(g(x)) where f(x) = x^2 - 9 and g(x) = 10 - x/2. The problem involves understanding the domain and range of individual functions and solving inequalities. Suitable for high school students in Grades 10-12.

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