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When learning about functions, it's crucial to understand what happens when a denominator in a function equals zero. In mathematics, division by zero is undefined because there is no number that multiplied by zero gives a nonzero number. This is why when you come across a function like \( f(x)=\frac{2x+1}{x-1} \), your first instinct should be to identify conditions that would make the denominator zero, as this would result in an undefined value for the function.

In the given exercise, inspecting the denominator \( x-1 \) and setting it to zero, as seen in Step 1 and Step 2 of the solution, provides the critical information: \( x=1 \) is the problematic value. Every other real number can be safely used in the function without causing a mathematical error. This little investigation helps us to avoid undefined function values, leading to a correct determination of the domain.

Why should we bother about undefined function values? In algebra, a function is considered a special kind of relation where every input has a unique output. If a function gives no output or an ambiguous result, it fails to meet this definition and is therefore not valid at that point. In the context of our practice problem \( f(x)=\frac{2x+1}{x-1} \), the function becomes undefined when the denominator is zero, which happens when \( x=1 \).

This singular value is excluded from the domain of the function, ensuring the function's integrity across its domain. Excluding undefined values is like careful gardening; removing one bad weed could save the entire garden. The process ensures all values within the domain yield a meaningful output, establishing a clear set of legitimate input values.

Interval notation is a system in mathematics used to describe the domain or range of a function that encompasses all the numbers between two endpoints. It is a concise way to express intervals without writing lengthy descriptions. This notation utilizes parentheses \(( )\) for open intervals and brackets \([ ]\) for closed intervals. An open interval does not include the endpoints, while a closed interval does.

In the final step of our solution, interval notation comes into play to describe all the real numbers that can be inputs for the function except for the excluded value at which the function is undefined. The domain for \( f(x) \) is expressed as \( x \in (-\infty, 1) \cup (1, \infty) \). This notation clearly indicates that the function exists for all real numbers except for the point at \( x=1 \), which is left out due to the risk of division by zero. Interval notation is an efficient and universally understood method to represent such sets of numbers.