In Exercises 1–8, find the domain of each rational function. f(x)=5x/(x−4)

Verified step by step guidance

1. Identify the rational function given: $f\left(x\right)=\frac{5x}{x-4}$.
2. Recall that the domain of a rational function includes all real numbers except where the denominator is zero, because division by zero is undefined.
3. Set the denominator equal to zero to find the values to exclude: $x-4=0$.
4. Solve the equation for $x$: $x=4$.
5. Conclude that the domain of $f(x)$ is all real numbers except $x\; =\; 4$, which can be written in interval notation as $(-\infty ,\; 4)\; \cup \; (4,\; \infty )$.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding rational functions involves recognizing that the function is undefined where the denominator Q(x) equals zero, which affects the domain.

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 rational functions, the domain excludes values that make the denominator zero, since division by zero is undefined.

Solving for Restrictions on the Domain

To find the domain of a rational function, set the denominator equal to zero and solve for x. These values are excluded from the domain. For example, if the denominator is (x - 4), then x = 4 is excluded, so the domain is all real numbers except 4.