$$ g\left(x\right)=\sqrt{2x+6} $$$$ h\left(x\right)=x-1 $$find the domain and composition of $$ g\left(h\left(x\right)\right) $$

To find the domain and composition of $ g(h(x)) $, where $ g(x) = \sqrt{2x + 6} $ and $ h(x) = x - 1 $, we follow these steps:

1. **Composition of the Functions:**
   We first find $ g(h(x)) $.

   Given $ h(x) = x - 1 $, we substitute $ h(x) $ into $ g(x) $:
   \[
   g(h(x)) = g(x - 1) = \sqrt{2(x - 1) + 6}
   \]
   Simplify inside the square root:
   \[
   g(x - 1) = \sqrt{2x - 2 + 6} = \sqrt{2x + 4}
   \]

2. **Domain of the Composition:**
   To determine the domain of $ g(h(x)) $, we need to ensure that the expression inside the square root is non-negative:
   \[
   2x + 4 \geq 0
   \]
   Solving this inequality:
   \[
   2x + 4 \geq 0 \implies 2x \geq -4 \implies x \geq -2
   \]

   Therefore, the domain of $ g(h(x)) $ is $ x \geq -2 $.

**Summary:**
- The composition $ g(h(x)) = \sqrt{2x + 4} $.
- The domain of $ g(h(x)) $ is $ x \geq -2 $.

Would you like more details or have any questions?

Here are 8 related questions for further exploration:

1. How would you find the domain of $ h(g(x)) $ if $ h $ and $ g $ were composed in reverse?
2. What are the ranges of $ g(x) $ and $ h(x) $?
3. How do you determine the range of $ g(h(x)) $?
4. What happens if the inside of the square root in $ g(x) $ is negative?
5. Can you graph the functions $ g(x) $ and $ h(x) $ and their composition?
6. How do you verify the domain of a composite function algebraically?
7. What are some real-world applications of composite functions?
8. How do you find the inverse of a composite function?

**Tip:** When working with composite functions, always ensure that the output of the inner function falls within the domain of the outer function to correctly determine the domain of the composition.

Explore the domain and composition of functions g(h(x)) where g(x) = √(2x + 6) and h(x) = x - 1. Understand how to determine the domain of g(h(x)) and the steps involved in function composition. Suitable for advanced high school students.

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