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Domain and Range

As we’ve seen earlier, functions involve a dependent and independent variable, and the two variables have a relationship. For a series of numbers or values of the independent variable, we can find numbers or values of the dependent variable. These numbers or values can be put in a table, and then plotted to obtain a graph.

Before we do that, let’s define two important aspects of a function – viz. domain and range.

Domain is a set of all real numbers for x (the independent variable) for which a function is defined.

Range is a set of all real values for y or f(x) – the dependent variable, as the values of x varies.

For example, the domain and range for the function y = f(x) = x2 is:

Domain: x takes all real values/numbers (both positive and negative numbers)

Range :  y = f(x) = x2 is always positive or equal to zero. So y =  f(x) ~>=~ 0

We represent domain in two ways:

  1. Domain : {all real x}, and is it read as “Domain is the set of all real numbers”, or
  2. Domain : {x |  x~ in~ R }, and it is read as “Domain is a set of all x such that x belongs to real numbers”. Sometimes ‘such that’ (the symbol |) is also represented by : (colon).

Similarly, we can represent range in two ways, and they are both read as “Range is a set of y such that  y~>=~0” :

  1. Range : {y:  y~>=~0 }, or
  2. Range : {y |  y~>=~0 }.

Now let us look at a few examples of finding domain and range for any given function.