How to determine the domain and range of the functions?

Definition of the Domain of a Function
For a function f defined by an expression with variable x, the implieddomain of f is the set of all real numbers variable x can take suchthat the expression defining the function is real. The domain can also be given explicitly.
Example: Find the domain of function f definedby
f (x) = 1 / ( x - 5)
x can take any real number except 5 since x = 5 would make the denominator equal to zero and the division by zero is not allowed in mathematics. Hence the domain in interval notation is given by $-\mathrm{\infty }<x<+\mathrm{\infty }$ but, x 5
Definition of the Range of a Function
The range of f is the set of all values that the function takes when x takes values in the domain.
Example: Find the range of function f defined by:
$f(x)=x^2-2$
The domain of this function is the set of all real numbers. The range is the set ofvalues that f(x) takes as x varies. If x is a real number,$x^2$ is either positive or zero. Hence we can write the following:
$x^2>=0$
Subtract -2 to both sides to obtain
$x^2-2>=-2$
The last inequality indicates that $x^2-2$ takes all values greater that or equal to -2. The range of f is given by
[ -2 ,+infinity)