The function f(x)=(x-1)/(x-4)*sqrt(x+2) is negative for x in...
a. (1,4)
b. (-infinity, 4)
c. (-infinity, 1)
d. (4, infinity)
To determine where a function is negative, we need to find the intervals where the function's values are less than zero.
Let's start analyzing the function f(x)=(x-1)/(x-4)*√(x+2).
First, note that the function is defined as long as the denominator is not equal to zero. So, x ≠ 4.
Next, let's focus on the numerator (x-1) and the denominator (x-4):
1. Numerator (x-1):
To find where the numerator is negative, we solve the inequality x - 1 < 0.
Adding 1 to both sides, we get: x < 1.
So, the numerator (x-1) is negative for x < 1.
2. Denominator (x-4):
To find where the denominator is negative, we solve the inequality x - 4 < 0.
Adding 4 to both sides, we get: x < 4.
Since the denominator cannot be equal to zero (x ≠ 4), we disregard this solution.
Now, let's analyze the expression √(x+2):
For this expression to be defined, x+2 must be greater than or equal to zero.
So, x+2 ≥ 0.
Subtracting 2 from both sides, we get: x ≥ -2.
Therefore, the expression √(x+2) is always positive or zero for x ≥ -2.
Now let's combine the results:
- The numerator (x-1) is negative for x < 1.
- The denominator (x-4) cannot be zero, so we disregard x = 4.
- The expression √(x+2) is always positive or zero for x ≥ -2.
Considering these conditions, the function f(x) is negative for x < 1.
Therefore, the correct option is:
c. (-infinity, 1)