Answer:
Positive interval(s): None
Negative interval(s): (-∞, 1) and (1, ∞)
Increasing interval(s): None
Decreasing interval(s): (-∞, 1) and (1, ∞)
Explanation:
To determine the positive interval(s), negative interval(s), increasing interval(s), and decreasing interval(s) of the function f(x) = (x - 3)/(x - 1), we need to analyze the sign and behavior of the function over different ranges of x. Let's break it down step by step:
Find the critical point(s) by setting the denominator equal to zero and solving for x:
x - 1 = 0
x = 1
We have a vertical asymptote at x = 1 since the function is undefined when the denominator is zero.
Determine the sign of the function in different intervals. We can use test points or consider the behavior of the numerator and denominator.
a) Interval (-∞, 1):
Choose x = 0 as a test point.
f(0) = (0 - 3)/(0 - 1) = 3/(-1) = -3
The function is negative in this interval.
b) Interval (1, ∞):
Choose x = 2 as a test point.
f(2) = (2 - 3)/(2 - 1) = -1/1 = -1
The function is negative in this interval as well.
Analyze the increasing and decreasing intervals by considering the behavior of the function around the critical point(s).
a) Interval (-∞, 1):
Since the function is negative in this interval and does not cross the x-axis, it is decreasing.
b) Interval (1, ∞):
Again, the function is negative in this interval and does not cross the x-axis, so it is also decreasing.