Determining the Domain of fx log(x / (x^2 - 1))

Understanding the domain of a function is a crucial step in its analysis, particularly for logarithmic functions. The function we are dealing with here is fx log(x / (x^2 - 1)). The domain of this function is defined by ensuring the argument of the logarithm, i.e., x / (x^2 - 1), is positive. This step-by-step guide will illustrate how to determine this domain.

Step 1: Identify When the Fraction is Positive

For the fraction x / (x^2 - 1) to be positive, the numerator and the denominator must both have the same sign.

Numerator: x Denominator: x^2 - 1 (x 1)(x - 1)

The denominator equals zero when x -1 or x 1, which are points we must exclude from our domain to avoid division by zero.

Step 2: Analyze the Denominator

The denominator x^2 - 1 can be zero if:

x 1 0 > x -1 x - 1 0 > x 1

These points divide the number line into intervals:

?∞ x -1 -1 x 1 1 x ∞

We will test the sign of the fraction in each interval to determine where it is positive.

Step 3: Test the Intervals

Interval: (?∞, -1) Choose x -2 -2 / (-2^2 - 1) -2 / (4 - 1) -2 / 3 0 **Negative** Interval: (-1, 1) Choose x 0 0 / (0^2 - 1) 0 / -1 0 **Zero (not positive)** Interval: (1, ∞) Choose x 2 2 / (2^2 - 1) 2 / (4 - 1) 2 / 3 0 **Positive**

Step 4: Combine Results

From the analysis:

The fraction is negative in (?∞, -1) The fraction is zero in (-1, 1) The fraction is positive in (1, ∞)

Hence, the fraction is positive in (1, ∞).

Step 5: Exclude Points Where the Denominator is Zero

The function is not defined at x -1 and x 1 because the denominator is zero at these points. Therefore, these points must be excluded from the domain.

Conclusion

The domain of fx log(x / (x^2 - 1)) is:

boxed{(1, ∞)}

Alternatively, you can also express the domain of fx as -1 x 0 U (1, ∞).