To find the domain of [tex]\((f \circ g)(x)\)[/tex], which represents [tex]\(f(g(x))\)[/tex], let's follow these steps:
1. Understand the domain of [tex]\(f(x)\)[/tex]:
The function [tex]\(f(x) = \frac{x-3}{x}\)[/tex] has a restriction due to the denominator: [tex]\(x \neq 0\)[/tex]. Hence, the domain of [tex]\(f(x)\)[/tex] is all real numbers except [tex]\(x = 0\)[/tex].
2. Apply [tex]\(g(x)\)[/tex]:
The function [tex]\(g(x) = 5x - 4\)[/tex] is a linear function and is defined for all real numbers. There are no restrictions on its domain.
3. Determine the domain of [tex]\(f(g(x))\)[/tex]:
We need to find when the argument [tex]\(g(x)\)[/tex] makes [tex]\(f(g(x))\)[/tex] undefined. This occurs if:
[tex]\[
g(x) = 0
\][/tex]
1. Solve for [tex]\(x\)[/tex]:
[tex]\[
5x - 4 = 0
\][/tex]
[tex]\[
5x = 4
\][/tex]
[tex]\[
x = \frac{4}{5}
\][/tex]
Therefore, [tex]\(x = \frac{4}{5}\)[/tex] is the value that makes [tex]\(g(x)\)[/tex] equal to zero, causing [tex]\(f(g(x))\)[/tex] to be undefined.
4. State the domain of [tex]\((f \circ g)(x)\)[/tex]:
The domain of [tex]\((f \circ g)(x)\)[/tex] includes all real numbers except the value where [tex]\(g(x)\)[/tex] turns the argument of [tex]\(f\)[/tex] into 0. Hence, we exclude [tex]\(x = \frac{4}{5}\)[/tex].
By analyzing the exclusions, the domain of [tex]\((f \circ g)(x)\)[/tex] is:
[tex]\[
\{x \mid x \neq \frac{4}{5}\}
\][/tex]
Therefore, the correct option is:
[tex]\(\left\{x \mid x \neq \frac{4}{5}\right\}\)[/tex].
