To find the domain of [tex]\((f \circ g)(x)\)[/tex], where [tex]\(f(x) = \frac{1}{x-2}\)[/tex] and [tex]\(g(x) = \sqrt{x+4}\)[/tex], we need to determine the set of all [tex]\(x\)[/tex] values for which the composite function [tex]\(f(g(x))\)[/tex] is defined.
### Step-by-Step Solution:
1. Identify the Domain of [tex]\(g(x)\)[/tex]:
- Since [tex]\(g(x) = \sqrt{x+4}\)[/tex], we need [tex]\(x + 4 \geq 0\)[/tex] to ensure that the square root is defined (it must be a non-negative number).
- Solving [tex]\(x + 4 \geq 0\)[/tex], we get [tex]\(x \geq -4\)[/tex].
- Therefore, the domain of [tex]\(g(x)\)[/tex] is [tex]\(x \geq -4\)[/tex], which can be written as [tex]\([-4, \infty)\)[/tex].
2. Find [tex]\(g(x)\)[/tex] Values that Ensure [tex]\(f(g(x))\)[/tex] is Defined:
- We must ensure [tex]\(g(x)\)[/tex] is in the domain of [tex]\(f(x) = \frac{1}{x - 2}\)[/tex].
- The function [tex]\(f(x)\)[/tex] is undefined when [tex]\(x = 2\)[/tex], as we cannot divide by zero.
- So, we need to find the values of [tex]\(x\)[/tex] for which [tex]\(g(x) \neq 2\)[/tex].
3. Set [tex]\(g(x) \neq 2\)[/tex]:
- Solve [tex]\(g(x) = 2\)[/tex], which gives [tex]\(\sqrt{x + 4} = 2\)[/tex].
- Square both sides: [tex]\((\sqrt{x + 4})^2 = 2^2\)[/tex], so we have [tex]\(x + 4 = 4\)[/tex].
- Solving [tex]\(x + 4 = 4\)[/tex] gives [tex]\(x = 0\)[/tex].
- Therefore, [tex]\(g(x) = 2\)[/tex] when [tex]\(x = 0\)[/tex], and at this point, [tex]\(f(g(x))\)[/tex] is undefined.
4. Combine the Conditions:
- [tex]\(x\)[/tex] must satisfy the condition [tex]\(x \geq -4\)[/tex], and at the same time, it must not be [tex]\(0\)[/tex] because [tex]\(f\)[/tex] will be undefined at [tex]\(x = 0\)[/tex].
- Hence, the domain of [tex]\((f \circ g)(x)\)[/tex] excludes [tex]\(0\)[/tex] and starts from [tex]\(-4\)[/tex] up to infinity.
5. Write the Final Domain in Interval Notation:
- This combination is expressed in interval notation as:
[tex]\[
[-4, 0) \cup (0, \infty)
\][/tex]
### Conclusion:
The domain of [tex]\((f \circ g)(x)\)[/tex] is [tex]\([ -4, 0) \cup (0, \infty)\)[/tex].
From the options provided, the correct answer is:
[tex]\[ [-4, 0) \cup (0, \infty) \][/tex]
