Answer :
To determine the domain of the composite function \((f \circ g)(x)\), we need to analyze the domains of the individual functions \(f(x)\) and \(g(x)\) and then find the domain for the composition.
Firstly, let's look at the individual functions:
1. \(f(x) = x + 7\):
- The function \(f(x)\) is defined for all real numbers \(x\). Thus, the domain of \(f(x)\) is \(\mathbb{R}\).
2. \(g(x) = \frac{1}{x - 13}\):
- The function \(g(x)\) involves a division, which means its denominator must not be equal to zero. Therefore, \(x - 13 \neq 0\) which implies \(x \neq 13\). Thus, the domain of \(g(x)\) is \(\mathbb{R} \setminus \{13\}\).
Now, let's consider the composite function \((f \circ g)(x)\):
- The composite function is defined as \((f \circ g)(x) = f(g(x))\).
- For \(f(g(x))\) to be defined, \(g(x)\) itself must first be well-defined. This means that \(x\) must belong to the domain of \(g(x)\).
Since \(g(x)\) is defined for all real numbers except \(x = 13\), the domain of \((f \circ g)(x)\) will inherit the restriction from the domain of \(g(x)\).
Therefore, the domain of \((f \circ g)(x)\) is all real numbers except \(x = 13\).
Thus, the domain of \((f \circ g)(x)\) is:
[tex]\[ \{x \mid x \neq 13\} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{\{x \mid x \neq 13\}} \][/tex]
Firstly, let's look at the individual functions:
1. \(f(x) = x + 7\):
- The function \(f(x)\) is defined for all real numbers \(x\). Thus, the domain of \(f(x)\) is \(\mathbb{R}\).
2. \(g(x) = \frac{1}{x - 13}\):
- The function \(g(x)\) involves a division, which means its denominator must not be equal to zero. Therefore, \(x - 13 \neq 0\) which implies \(x \neq 13\). Thus, the domain of \(g(x)\) is \(\mathbb{R} \setminus \{13\}\).
Now, let's consider the composite function \((f \circ g)(x)\):
- The composite function is defined as \((f \circ g)(x) = f(g(x))\).
- For \(f(g(x))\) to be defined, \(g(x)\) itself must first be well-defined. This means that \(x\) must belong to the domain of \(g(x)\).
Since \(g(x)\) is defined for all real numbers except \(x = 13\), the domain of \((f \circ g)(x)\) will inherit the restriction from the domain of \(g(x)\).
Therefore, the domain of \((f \circ g)(x)\) is all real numbers except \(x = 13\).
Thus, the domain of \((f \circ g)(x)\) is:
[tex]\[ \{x \mid x \neq 13\} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{\{x \mid x \neq 13\}} \][/tex]
