In Exercises 109–112, find the domain of each logarithmic function. f(x) = log[(x+1)/(x-5)]

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1. Recall that the domain of a logarithmic function \( f(x) = \log(g(x)) \) consists of all values of \( x \) for which the argument \( g(x) \) is positive, i.e., \( g(x) > 0 \).
2. Identify the argument of the logarithm in the function: \( \frac{x+1}{x-5} \). We need to find where \( \frac{x+1}{x-5} > 0 \).
3. Determine the critical points where the numerator or denominator is zero: \( x+1=0 \Rightarrow x=-1 \) and \( x-5=0 \Rightarrow x=5 \). These points divide the number line into intervals to test.
4. Test the sign of \( \frac{x+1}{x-5} \) in each interval: \( (-\infty, -1) \), \( (-1, 5) \), and \( (5, \infty) \). For each interval, pick a test value and check if the fraction is positive.
5. Combine the intervals where the fraction is positive, excluding points where the denominator is zero (since division by zero is undefined), to write the domain of \( f(x) \).

Key Concepts

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For logarithmic functions, the argument inside the log must be positive, so determining the domain involves finding all x-values that make the argument greater than zero.

Properties of Logarithmic Functions

A logarithmic function log_b(A) is defined only when the argument A is positive (A > 0). This means that for f(x) = log[(x+1)/(x-5)], the expression (x+1)/(x-5) must be greater than zero to ensure the function is valid.

Inequalities Involving Rational Expressions

To find where (x+1)/(x-5) > 0, one must solve a rational inequality. This involves determining the sign of the numerator and denominator, identifying critical points where the expression is zero or undefined, and testing intervals to find where the expression is positive.