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.