Friday, July 6, 2012

Domain and Range of a Function

In mathematics, a function is a relation between a set of inputs and a set of potential outputs with the property that each input is related to exactly one output. An example of such a relation is defined by the rule f(x) = x2, which relates an input x to its square, which are both real numbers. The output of the function f corresponding to an input x is denoted by f(x) (read "f of x"). If the input is –3, then the output is 9, and we may write    f(–3) = 9.

Thus we have two quantities (called "variables") and we observe that there is a relationship between them. If we find that for every value of the first variable there is only one value of the second variable, then we say: "The second variable is a function of the first variable."

The first variable is the independent variable (usually written as x), and the second variable is the dependent variable (usually written as y).

The domain of a function is the set of all possible x-values which will make the function work and will output real y-values. 

The range of a function is the possible y values of a function that result when we substitute all the possible x-values into the function.

The following applet shows the graph of sine function. Its domain ( which can be tracked by point B) and range(which can be tracked by point C) are respectively (-∞ , ∞) and [-1,1].

This is a Java Applet created using GeoGebra from - it looks like you don't have Java installed, please go to


  1. This is nice! I made a similar application last fall for my students to practice domain and range. I'm not completely happy with it because it doesn't show closed vs. open intervals, but here it is:

  2. Thank you for visiting and the link.

  3. Functions are very necessary to learn differentiation and integration.The whole limit theory and calculus depend on functions.

  4. sir , why the range lie between only [-1,1] ??

  5. because 'archana' the limits of a sine or cosine waveform are -1 to +1.
    If we had chosen f(x) = 2x/(x-3)
    we have a limitation in the domain, the values that x can take, as when x = 3 the denominator = 0 and the function f(x) becomes infinite in value. We would show this as an asymptote on the graph.
    The Range of that function f(x) can be found by letting another variable, say y, equal f(x) and then re-arranging to make x the subject.
    In this case we would get x = 3y/(y-2)
    We call this the inverse of the function

    f^-1 (x) = 3x/(x-2)

    In this case the limiting value would be x = 2 and this is the limiting value in the range or values that the function f(x) can take.

    In summary for the function chosen we have a domain:
    x can take all values except x = 3
    and a range:
    f(x) can take all values except f(x) = 2