CLASS-11
RELATION & FUNCTION - FURTHER DISCUSSION ON DOMAIN OF A FUNCTION

Further Discussion On The Domain Of A Function –

Consider the following functions.

                      1

1.) The function ------ has no meaning when x = 0, so this function is 

                      x

defined for all real numbers except 0.

2.) The function √x is defined for only the non-negative real numbers.

3.) Since a negative number does not have a real square root, therefore, the function √x(8 – x).

Is defined only when x(8 – x) ≥ 0.

This will be true only when both the factors x and (8 – x) are greater than or equal to 0.

If, x ≥ 0 and (8 – x) ≥ 0, then 0 ≤ x ≤ 8.

The alternative x ≤ 0 and (8 – x) ≤ 0, is impossible because (8 – x) ≤ 0                                                                                                        => 8 ≤ x  =>  x  ≥ 8.

We can’t have x ≤ 0 and x ≥ 8 at the same time. So the function √x(8 – x) is defined only if  0 ≤ x ≤ 8.

Definition – The set of numbers x for which function f(x) is defined is called the domain of the function.

Thus, the domain of the function f(x) = √x(8 – x) is the set of real numbers such that 0 ≤ x ≤ 8. These are any number for which √x(8 – x) exists.