To determine the domain of [tex]\((f \circ g)(x)\)[/tex], where [tex]\( f(x) = \frac{x - 3}{x} \)[/tex] and [tex]\( g(x) = 5x - 4 \)[/tex], we need to follow these steps:
1. Determine the domain of [tex]\( g(x) \)[/tex]:
- The function [tex]\( g(x) = 5x - 4 \)[/tex] is a linear function, and linear functions are defined for all real numbers. Hence, the domain of [tex]\( g(x) \)[/tex] is all real numbers, [tex]\( \mathbb{R} \)[/tex].
2. Determine the domain of [tex]\( f(x) \)[/tex]:
- The function [tex]\( f(x) = \frac{x - 3}{x} \)[/tex] is a rational function and is undefined where the denominator is zero.
- The denominator [tex]\( x \)[/tex] is zero when [tex]\( x = 0 \)[/tex]. Therefore, [tex]\( f(x) \)[/tex] is undefined at [tex]\( x = 0 \)[/tex], and the domain of [tex]\( f(x) \)[/tex] is [tex]\( \{ x \mid x \neq 0 \} \)[/tex].
3. Compose [tex]\( f \)[/tex] and [tex]\( g \)[/tex] to find [tex]\( (f \circ g)(x) \)[/tex]:
- The composite function [tex]\( (f \circ g)(x) = f(g(x)) \)[/tex] involves evaluating [tex]\( f \)[/tex] at [tex]\( g(x) \)[/tex].
- Substitute [tex]\( g(x) \)[/tex] in place of [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex]:
[tex]\[
f(g(x)) = f(5x - 4) = \frac{(5x - 4) - 3}{5x - 4} = \frac{5x - 7}{5x - 4}
\][/tex]
4. Determine where the composite function [tex]\( (f \circ g)(x) \)[/tex] is undefined:
- The composite function [tex]\( f(g(x)) \)[/tex] will be undefined where the denominator of [tex]\( f(g(x)) \)[/tex] equals zero.
- The denominator of [tex]\( f(g(x)) \)[/tex] is [tex]\( 5x - 4 \)[/tex]. Set this equal to zero to find the critical point:
[tex]\[
5x - 4 = 0
\][/tex]
[tex]\[
x = \frac{4}{5}
\][/tex]
Thus, the composite function [tex]\( (f \circ g)(x) \)[/tex] is undefined at [tex]\( x = \frac{4}{5} \)[/tex], and the domain of [tex]\( (f \circ g)(x) \)[/tex] excludes this value.
5. Conclusion:
- The domain of [tex]\( (f \circ g)(x) \)[/tex] is all real numbers except [tex]\( x = \frac{4}{5} \)[/tex]. Therefore, the correct domain is:
[tex]\[
\left\{ x \left\lvert x \neq \frac{4}{5} \right.\right\}
\][/tex]
Thus, the answer is:
[tex]\[
\boxed{\left\{ x \mid x \neq \frac{4}{5} \right\}}
\][/tex]
