Answer :
To find the domain of the function f(x) = log | (3x-5) / (x-1) |, we need to consider the restrictions on the logarithmic function. The domain of a logarithmic function is the set of all real numbers that make the argument inside the logarithm function positive.
1. Set the argument inside the logarithm function greater than zero to find the domain:
(3x - 5) / (x - 1) > 0
2. Find the critical points by setting the numerator and denominator equal to zero:
3x - 5 = 0 and x - 1 = 0
Solving these equations gives x = 5/3 and x = 1.
3. Create intervals using these critical points and test points within each interval:
I. Test a point x < 1, say x = 0:
(3(0) - 5) / (0 - 1) = -5 / -1 = 5, which is positive.
II. Test a point 1 < x < 5/3, say x = 2:
(3(2) - 5) / (2 - 1) = 1 / 1 = 1, which is positive.
III. Test a point x > 5/3, say x = 2:
(3(2) - 5) / (2 - 1) = 1 / 1 = 1, which is positive.
4. Based on the test results, the domain of the function f(x) = log | (3x-5) / (x-1) | is:
x ∈ (-∞, 1) U (1, 5/3) U (5/3, ∞)
Therefore, the domain of the function f(x) is all real numbers except x = 1.
