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Interval notation is a mathematical system to describe sets of numbers along a number line. It comes in handy when you want to specify the range of possible values that a variable, like the independent variable of a function, can take on. For instance, to indicate that a variable 'x' can be any number between 1 and 5, we use the interval notation: \((1, 5)\). This tells us that 'x' can be greater than 1 and less than 5, but not exactly equal to these boundary numbers.
In cases where 'x' can be equal to the endpoints, we replace parentheses with brackets. So, \([1, 5]\) includes 1 and 5 as possible values for 'x'. Another aspect of interval notation involves infinity. When the set of numbers extends indefinitely in either direction, we use the symbols \(-\infty\) for negative infinity and \(\infty\) for positive infinity. To depict all real numbers, we write \((-\infty, \infty)\), showing that there are no bounds – 'x' can be any real number. When using interval notation, it's crucial to use the correct symbols to communicate the precise range accurately and concisely.

In mathematics, an expression involving a fraction isn't defined when the denominator equates to zero, as division by zero is undefined. For functions that can be written as a ratio of two expressions, identifying where the denominator equals zero is essential in determining the function's domain.
To illustrate, if we have a function like \(f(x) = \frac{1}{x^2 + 1}\), we need to ensure that the denominator \(x^2 + 1\) never equals zero because that would make the function undefined. To find these values, we set the denominator equal to zero and solve for 'x'. If there are real-number solutions to this equation, those would be excluded from the function's domain. However, if there are no real-number solutions – for example, if solving the equation leads to the square root of a negative number – then the denominator will never be zero for any real 'x', and thus, the function has a domain of all real numbers.

Real numbers encompass all the numbers on the number line, including all positive and negative integers, fractions, and irrational numbers. They offer a representation for any quantity that can be measured along the continuum of the number line. Discrete quantities, such as counting numbers, and continuous quantities, like measurements, can all be described using real numbers.
Understanding the concept of real numbers is crucial when examining the domain of a function, which is the complete set of 'input' values. When a function's domain is 'all real numbers', it means that you can substitute any real number into the function and obtain a valid output. Mathematically, this is often notated as \((-\infty, \infty)\) in interval notation, encompassing the entirety of the number line without restriction (assuming the function has no other limitations, such as a square root of a negative number or a zero denominator issue).