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The domain of a function is a set that includes all the possible input values, which will give an output when applied to the function. In simpler terms, it's where the function "lives," including every value you can use for the input that won't break the function.
For a rational function like the one in our exercise, specifically \( k(r) = \frac{r}{5r + 2} \), the domain is all real numbers except the ones that make the denominator zero. That's because division by zero is undefined in mathematics.
To identify the domain, you follow these steps:

• Focus on the denominator: Any expression or number that makes the denominator zero must be excluded from the domain.
• Solve for these restricted values, as seen in the step-by-step solution where \(5r + 2 = 0\) is solved for \(r\).
• Exclude these values from the collection of all real numbers, as they make the function undefined.

Understanding domains help ensure you're only working with valid and useful values that keep the function working properly.

Interval notation is a way of writing the domain (or any set of numbers) concisely and clearly. It's especially useful in mathematics when you talk about continuous blocks of numbers or when certain values are excluded.
For our rational function \( k(r) = \frac{r}{5r + 2} \), using interval notation allows us to express where the function is defined. Since we found the function is undefined at \(r = -\frac{2}{5}\), we exclude this point:

• The notation \( (-\infty, -\frac{2}{5}) \cup (-\frac{2}{5}, \infty) \) tells us the domain covers all real numbers, except \(-\frac{2}{5}\).
• Parentheses \((\) indicate that the endpoints are not included in the domains (also known as open intervals).
• The union symbol \(\cup\) combines separate intervals into a single domain.

This notation is a useful tool to clearly communicate the domain without needing lengthy explanations.

In mathematics, a function is said to be "undefined" when it does not produce a valid result for a specific input. This can happen for several reasons, but most commonly with rational functions, it's because the denominator is zero. Dividing by zero is undefined as it does not result in a finite number or suitable outcome.
To determine when a function is undefined, set the denominator equal to zero and solve. This point where the denominator equals zero is excluded from the function’s domain.
For instance, with \( k(r) = \frac{r}{5r + 2} \), setting \(5r + 2 = 0\) reveals \(r = -\frac{2}{5}\) as the point where the function is undefined.

• Undefined values are either because of division by zero or other constraints, like square roots of negative numbers in real numbers.
• It's crucial to identify these points to avoid errors in calculations and describe the function accurately.

Understanding where a function is undefined is important in graphing functions, as these points often correspond to vertical asymptotes or holes in the graph.