The domain of a function is not always all real numbers. The domain consists of all the input values (usually x-values) for which the function is defined. In many cases, certain mathematical operations, such as division by zero or taking the square root of a negative number, can restrict the set of input values.
For example, consider the function f(x) = 1/(x – 2). In this case, the function is undefined when x = 2 because you cannot divide by zero. Therefore, the domain of this function is all real numbers except x = 2, which can be expressed in interval notation as (−∞, 2) ∪ (2, +∞).
Another example is the square root function, g(x) = √(x). The square root is defined only for non-negative numbers, so the domain of this function is [0, +∞).
In conclusion, while some functions do have a domain of all real numbers, many functions have restricted domains based on their mathematical definitions. It’s essential to analyze each function individually to determine its specific domain.