To determine the domain of [tex]\( (f \circ g)(x) \)[/tex], we will follow these steps:
1. Understand the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- [tex]\( f(x) = \frac{x-3}{x} \)[/tex]
- [tex]\( g(x) = 5x - 4 \)[/tex]
2. Find the composition [tex]\( f(g(x)) \)[/tex]:
- Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex].
- This gives us [tex]\( f(g(x)) = f(5x - 4) \)[/tex].
- Replace [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] with [tex]\( 5x - 4 \)[/tex], resulting in [tex]\( f(5x - 4) = \frac{(5x - 4) - 3}{5x - 4} \)[/tex].
- Simplify the numerator: [tex]\( (5x - 4) - 3 = 5x - 7 \)[/tex].
- Therefore, [tex]\( f(g(x)) = \frac{5x - 7}{5x - 4} \)[/tex].
3. Determine conditions for the domain:
- For the composition [tex]\( f(g(x)) \)[/tex] to be defined, we need the denominator to be non-zero.
- The denominator of [tex]\( f(g(x)) \)[/tex] is [tex]\( 5x - 4 \)[/tex].
- So, set [tex]\( 5x - 4 \neq 0 \)[/tex] which ensures the denominator is not zero.
- Solve the inequality: [tex]\( 5x - 4 \neq 0 \)[/tex], leading to [tex]\( x \neq \frac{4}{5} \)[/tex].
Thus, the domain of [tex]\( (f \circ g)(x) \)[/tex] is all real numbers [tex]\( x \)[/tex] such that [tex]\( x \ne \frac{4}{5} \)[/tex].
Therefore, the correct domain of [tex]\( (f \circ g)(x) \)[/tex] is:
[tex]\[
\left\{x \left\lvert\, x \neq \frac{4}{5}\right.\right\}
\][/tex]
So, the correct choice is:
[tex]\[
\left\{x \left\lvert\, x \neq \frac{4}{5}\right.\right\}
\][/tex]
