Given [tex]\( m(x)=\frac{x+5}{x-1} \)[/tex] and [tex]\( n(x)=x-3 \)[/tex], which function has the same domain as [tex]\( (m \circ n)(x) \)[/tex]?

A. [tex]\( f(x) = \frac{x+5}{x+1} \)[/tex]
B. [tex]\( g(x) = \frac{x-3}{x-1} \)[/tex]
C. [tex]\( h(x) = \frac{x-2}{x-4} \)[/tex]
D. [tex]\( k(x) = \frac{x-3}{x-2} \)[/tex]



Answer :

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.

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