Examples
Let f be a function on the real numbers defined by . This function takes as input any real number and outputs a real number two times the input. In this case, the codomain and the image are the same (i.e., the function is a surjection), so the range is unambiguous; it is the set of all real numbers.
In contrast, consider the function defined by . If the word "range" is used in the first sense given above, we would say the range of f is the codomain, all real numbers; but since the output of the sine function is always between -1 and 1, "range" in the second sense would say the range is the image, the closed interval from -1 to 1.
Read more about this topic: Range (mathematics)
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