To find the domain of the composite function [tex]\( (f \circ g)(x) \)[/tex], we need to ensure that [tex]\( g(x) \)[/tex] is within the domain of [tex]\( f(x) \)[/tex] and also that the denominators of both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are non-zero.
Let's start by examining the function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \frac{1}{-10x + 7} \][/tex]
The denominator must not be zero. Thus, we set:
[tex]\[ -10x + 7 \neq 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ -10x + 7 = 0 \][/tex]
[tex]\[ 10x = 7 \][/tex]
[tex]\[ x = \frac{7}{10} \][/tex]
So, [tex]\( x = \frac{7}{10} \)[/tex] is not in the domain of [tex]\( g(x) \)[/tex].
Next, we consider [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(x) = \frac{1}{-4x + 1} \][/tex]
We substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = \frac{1}{-4 \left( \frac{1}{-10x + 7} \right) + 1} \][/tex]
Again, we need to ensure that the denominator does not become zero:
[tex]\[ -4 \left( \frac{1}{-10x + 7} \right) + 1 \neq 0 \][/tex]
Solving for [tex]\( x \)[/tex], we get:
[tex]\[ -4 \left( \frac{1}{-10x + 7} \right) + 1 = 0 \][/tex]
First, isolate the fraction:
[tex]\[ -4 \left( \frac{1}{-10x + 7} \right) = -1 \][/tex]
[tex]\[ \left( \frac{1}{-10x + 7} \right) = \frac{1}{4} \][/tex]
Next, cross multiply:
[tex]\[ 4 = -10x + 7 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ -10x = 4 - 7 \][/tex]
[tex]\[ -10x = -3 \][/tex]
[tex]\[ x = \frac{3}{10} \][/tex]
So, [tex]\( x = \frac{3}{10} \)[/tex] must also be excluded from the domain of [tex]\( (f \circ g)(x) \)[/tex].
Therefore, the correct domain of [tex]\( (f \circ g)(x) \)[/tex] is all real numbers except [tex]\( x = \frac{7}{10} \)[/tex] and [tex]\( x = \frac{3}{10} \)[/tex].
Thus, the correct answer is:
All real numbers except [tex]\(\frac{7}{10}\)[/tex] and [tex]\(\frac{3}{10}\)[/tex].
