Sure, let's find the domain of [tex]\((f \circ g)(x)\)[/tex] for the functions [tex]\(f(x)=\frac{x-3}{x}\)[/tex] and [tex]\(g(x)=5x-4\)[/tex].
1. Understand the functions:
- [tex]\( f(x) = \frac{x - 3}{x} \)[/tex]
- [tex]\( g(x) = 5x - 4 \)[/tex]
2. Composite function [tex]\((f \circ g)(x)\)[/tex]:
To form [tex]\((f \circ g)(x)\)[/tex], we need to compute [tex]\( f(g(x)) \)[/tex].
- First, compute [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = 5x - 4
\][/tex]
- Next, substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[
f(g(x)) = f(5x - 4) = \frac{(5x - 4) - 3}{5x - 4} = \frac{5x - 7}{5x - 4}
\][/tex]
3. Determine the domain of [tex]\( f(g(x)) \)[/tex]:
- For [tex]\( f(g(x)) \)[/tex] to be defined, the denominator must not be zero.
- The denominator of [tex]\( f(g(x)) \)[/tex] is [tex]\( 5x - 4 \)[/tex].
4. Set the denominator not equal to zero:
[tex]\[
5x - 4 \neq 0
\][/tex]
[tex]\[
5x \neq 4
\][/tex]
[tex]\[
x \neq \frac{4}{5}
\][/tex]
The domain of [tex]\((f \circ g)(x)\)[/tex] is all real numbers [tex]\( x \)[/tex] except [tex]\( \frac{4}{5} \)[/tex].
Therefore, the answer is:
[tex]\[
\left\{x \left\lvert\, x \neq \frac{4}{5}\right.\right\}
\][/tex]
