To find the domain of the composite function [tex]\( (f \circ g)(x) \)[/tex], we need to consider the domains of both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
1. Find the domain of [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = \sqrt{2x + 4}
\][/tex]
For [tex]\( g(x) \)[/tex] to be defined, the expression inside the square root must be non-negative:
[tex]\[
2x + 4 \geq 0
\][/tex]
Solving this inequality:
[tex]\[
2x \geq -4
\][/tex]
[tex]\[
x \geq -2
\][/tex]
Thus, the domain of [tex]\( g(x) \)[/tex] is [tex]\( x \geq -2 \)[/tex].
2. Find the domain of [tex]\( f(g(x)) \)[/tex]:
[tex]\[
f(g(x)) = f(\sqrt{2x + 4}) = \frac{1}{7\sqrt{2x + 4} - 14}
\][/tex]
For [tex]\( f(g(x)) \)[/tex] to be defined, the denominator must not be zero:
[tex]\[
7\sqrt{2x + 4} - 14 \neq 0
\][/tex]
Solving for when this is zero:
[tex]\[
7\sqrt{2x + 4} = 14
\][/tex]
[tex]\[
\sqrt{2x + 4} = 2
\][/tex]
Squaring both sides:
[tex]\[
2x + 4 = 4
\][/tex]
[tex]\[
2x = 0
\][/tex]
[tex]\[
x = 0
\][/tex]
Therefore, [tex]\( x = 0 \)[/tex] must be excluded from the domain.
3. Combine the conditions:
The domain of [tex]\( g(x) \)[/tex] is [tex]\( x \geq -2 \)[/tex], but we also have to exclude the value where the denominator of [tex]\( f(g(x)) \)[/tex] becomes zero, which is [tex]\( x = 0 \)[/tex].
Hence, the domain of [tex]\( (f \circ g)(x) \)[/tex] is:
[tex]\[
\text{All real numbers } x \geq -2 \text{ other than 0}
\][/tex]
Thus, the correct answer is:
All real numbers [tex]\( x \geq -2 \)[/tex] other than 0.
