In Exercises 1–30, find the domain of each function. f(x) = 1/[3/(x - 1) - 2]

Verified step by step guidance

1. Identify the expression in the denominator: \( \frac{3}{x - 1} - 2 \).
2. Set the denominator equal to zero to find the values that make the function undefined: \( \frac{3}{x - 1} - 2 = 0 \).
3. Solve the equation \( \frac{3}{x - 1} = 2 \) to find the critical value of \( x \).
4. Multiply both sides by \( x - 1 \) to eliminate the fraction: \( 3 = 2(x - 1) \).
5. Solve for \( x \) to find the value that makes the denominator zero, and exclude this value from the domain.

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For rational functions, the domain is typically restricted by values that would make the denominator zero, as division by zero is undefined.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the case of f(x) = 1/[3/(x - 1) - 2], the function involves a rational expression, and understanding how to manipulate and simplify these expressions is crucial for finding the domain.

Finding Restrictions

To find the domain of a function, one must identify any restrictions on the variable. For the given function, we need to determine when the denominator equals zero, as these values will be excluded from the domain. This involves solving the equation set by the denominator.