Answer:
The domain of a function is the set of all possible values that the independent variable (usually denoted as x) can take. In this case, we need to determine which function, either m(x) or n(x), has the same domain as the composite function (m • n)(x).
To find the domain of the composite function (m • n)(x), we need to determine the values of x that make both m(x) and n(x) defined.
Let's start by finding the domain of m(x):
m(x) = (x + 5)/(x - 1)
The only value of x that would make m(x) undefined is when the denominator (x - 1) equals zero. So, we set x - 1 = 0 and solve for x:
x - 1 = 0
x = 1
Therefore, the function m(x) is undefined at x = 1.
Now let's find the domain of n(x):
n(x) = x - 3
Since n(x) is a linear function, it is defined for all real numbers. There are no restrictions on the domain of n(x).
Now, we need to find the function (m • n)(x), which represents the composite of m(x) and n(x):
(m • n)(x) = m(n(x))
Substituting n(x) into m(x), we get:
(m • n)(x) = m(x - 3) = ((x - 3) + 5)/((x - 3) - 1)
Simplifying further:
(m • n)(x) = (x + 2)/(x - 4)
The function (m • n)(x) is defined for all real numbers except for the values that make the denominator (x - 4) equal to zero. So, we set x - 4 = 0 and solve for x:
x - 4 = 0
x = 4
Therefore, the function (m • n)(x) is undefined at x = 4.
In summary, the function n(x) has the same domain as the composite function (m • n)(x). The domain of both functions is all real numbers, except for x = 4.