To solve this problem, we need to determine the domain of the function [tex]\((m \circ n)(x)\)[/tex], which means we need to evaluate [tex]\(m(n(x))\)[/tex] and identify any restrictions on [tex]\(x\)[/tex].
### Step-by-Step Solution:
1. Define the functions:
- [tex]\(m(x) = \frac{x+5}{x-1}\)[/tex]
- [tex]\(n(x) = x-3\)[/tex]
2. Find the composition [tex]\(m(n(x))\)[/tex]:
Substituting [tex]\(n(x) = x-3\)[/tex] into [tex]\(m(x)\)[/tex]:
[tex]\[
m(n(x)) = m(x-3) = \frac{(x-3) + 5}{(x-3) - 1}
\][/tex]
3. Simplify the composed function:
Simplify both the numerator and the denominator:
[tex]\[
m(n(x)) = \frac{x-3+5}{x-3-1} = \frac{x+2}{x-4}
\][/tex]
4. Determine the domain of [tex]\((m \circ n)(x)\)[/tex]:
The function [tex]\(m(n(x)) = \frac{x+2}{x-4}\)[/tex] will be undefined when its denominator is zero:
[tex]\[
x-4 = 0 \implies x = 4
\][/tex]
Therefore, [tex]\(x = 4\)[/tex] is the value where the composed function [tex]\((m \circ n)(x)\)[/tex] is undefined.
5. Domain of [tex]\((m \circ n)(x)\)[/tex]:
The domain of [tex]\((m \circ n)(x)\)[/tex] includes all real numbers except [tex]\(x = 4\)[/tex]. In interval notation, this is:
[tex]\[
\text{Domain of } (m \circ n)(x) = (-\infty, 4) \cup (4, \infty)
\][/tex]
6. Determine which function has the same domain:
For a function to have the same domain as [tex]\( (m \circ n)(x) = \frac{x+2}{x-4} \)[/tex], it must also be undefined at [tex]\(x = 4\)[/tex] only.
Let's analyze each function individually:
- [tex]\(m(x) = \frac{x+5}{x-1}\)[/tex]:
The function [tex]\(m(x)\)[/tex] is undefined when [tex]\(x - 1 = 0 \implies x = 1\)[/tex]. Therefore, the domain of [tex]\(m(x)\)[/tex] is [tex]\(\mathbb{R} \setminus \{1\}\)[/tex].
- [tex]\(n(x) = x-3\)[/tex]:
The function [tex]\(n(x)\)[/tex] is a linear polynomial and is defined for all real numbers. Therefore, its domain is [tex]\(\mathbb{R}\)[/tex].
- [tex]\((m \circ n)(x) = \frac{x+2}{x-4}\)[/tex]:
As we found, the domain is [tex]\(\mathbb{R} \setminus \{4\}\)[/tex].
Given this analysis, the only function with the same domain as [tex]\( (m \circ n)(x) \)[/tex] is [tex]\((m \circ n)(x)\)[/tex].
So, the function [tex]\( (m \circ n)(x) = \frac{x+2}{x-4} \)[/tex] itself has the same domain as the composition.
