Find the domain of each rational expression. (3x + 7) / (4x + 2)(x - 12)

Verified step by step guidance

1. Identify the rational expression: $\frac{3x+7}{(4x+2)(x-12)}$.
2. Recognize that the domain of a rational expression is all real numbers except where the denominator is zero.
3. Set the denominator equal to zero to find the values that are not in the domain: $(4x+2)(x-12)=0$.
4. Solve each factor for zero: $4x+2=0$ and $x-12=0$.
5. Determine the values of $x$ that make each factor zero, and exclude these from the domain.

Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding rational expressions is crucial because they can have restrictions on their values, particularly when the denominator equals zero, which leads to undefined expressions.

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 expressions, the domain excludes any values that make the denominator zero, as these would result in undefined expressions.

Finding Restrictions

To find the domain of a rational expression, one must identify the values of x that make the denominator equal to zero. This involves solving the equation formed by the denominator and excluding these values from the domain, ensuring that the expression remains valid.