To find the domain of the composite function [tex]\( (f \circ g)(x) \)[/tex], follow these steps:
1. Express the Composite Function:
Given:
[tex]\[
f(x) = \frac{x-3}{x}
\][/tex]
[tex]\[
g(x) = 5x - 4
\][/tex]
The composite function [tex]\( (f \circ g)(x) \)[/tex] is found by substituting [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[
(f \circ g)(x) = f(g(x)) = f(5x - 4)
\][/tex]
Substituting [tex]\( 5x - 4 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( f \)[/tex]:
[tex]\[
f(5x - 4) = \frac{(5x - 4) - 3}{5x - 4} = \frac{5x - 7}{5x - 4}
\][/tex]
2. Determine the Domain of [tex]\( g(x) \)[/tex]:
The function [tex]\( g(x) = 5x - 4 \)[/tex] is defined for all real numbers, so the domain of [tex]\( g(x) \)[/tex] is:
[tex]\[
(-\infty, \infty)
\][/tex]
3. Find the Domain Restrictions Imposed by [tex]\( f(x) \)[/tex]:
The function [tex]\( f(x) = \frac{x-3}{x} \)[/tex] requires that the denominator [tex]\( x \neq 0 \)[/tex]. When substituting [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex], we require:
[tex]\[
5x - 4 \neq 0
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
5x - 4 = 0 \implies x = \frac{4}{5}
\][/tex]
Therefore, [tex]\( x \)[/tex] cannot be [tex]\( \frac{4}{5} \)[/tex] because it would make the denominator zero in the composite function.
4. Combine the Domain Restrictions:
The domain of [tex]\( g(x) \)[/tex] is all real numbers, but since [tex]\( 5x - 4 \)[/tex] cannot equal zero, we exclude [tex]\( x = \frac{4}{5} \)[/tex] from the domain. Hence, the domain of [tex]\( (f \circ g)(x) \)[/tex] is:
[tex]\[
(-\infty, \frac{4}{5}) \cup (\frac{4}{5}, \infty)
\][/tex]
Thus, the domain of the composite function [tex]\( (f \circ g)(x) \)[/tex] is:
[tex]\[
\boxed{(-\infty, \frac{4}{5}) \cup (\frac{4}{5}, \infty)}
\][/tex]
