Thursday, 23 February 2012

Domain and Range of a Function

When functions are first introduced the terms domain and range of a function may have been briefly mentioned but never explained in full detail. The domain and range of a function are very important in a variety of subjects and you will definitely encounter it at some stage when maths begins to get more in depth.

The notation for a function, f with a domain of X and a range of Y is; f : X  Y. This however, still means nothing to you thought as you do not know what a domain or a range actually is, but once I explain it will all become clear.

The domain of a function is the values that can be put into the function whilst still producing a valid answer, basically this is all the values that x can take on. For example the function, f(x) = 1/x can have any value for x at all, that is not a 0. Mathematically speaking this is "all x ≠ 0". The range of a function are the possible values that can be produced by the function, the "range" of values that y can take on.

To combine everything we have done so far I will use another example. Find the domain and range of f(x) = √(x+3). The domain of this function is everything that makes what is inside the square root greater than or equal to 0, so for x ≥ -3 and the range of the function is everything that a square root can be, so everything positive which is y ≥ 0. Or, to put this mathematically:

And that is all there is to it! Domains and ranges are very easy to work out and deal with and they are very useful when it comes to nearly all areas of maths when graphs are a necessity.


  1. Cool! So then, would it be correct to say that the function f in your last example, is "surjective" or "onto"? Why or why not?

    1. A surjective function is one where every element of Y has an element in X. So basically every element in the range corresponds to one in the domain. And that is true, there is no value in the range that is not achieved from applying the function to an element in the domain.

      Thanks for the comment!

    2. I see, thanks for the response. So can you give an example of a non-surjective function?

    3. It depends how you define your range. If it is properly constructed then it will always be surjective. But if for example in my example I put, "f: {R ≥ -3} → {R}" it would be non-surjective as not all y values have a corresponding x value.