The domain of a function is the complete set of possible values of the independent variable in the function.
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.
When finding the domain, remember:
- The denominator (bottom) of a fraction cannot be zero
- The values under a square root and any even root sign must be positive
The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually) of a function, after we have substituted the domain 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.
When finding the range, remember:
- Substitute different x-values into the expression for y to see what is happening. (Is y always positive? Always negative? Or maybe not equal to certain values?)
- Make sure you look for minimum and maximum values of y.
Example
What are the domain and range of 
The domain of f is the set of all values of x for which the formula
is defined. This formula make sense if x is zero or a positive number. But if x is negative, f is not defined, the square root of negative number is not a real number. Therefore, the domain of is all nonnegative numbers x.
What is the range of f ? A square root is always nonnegative, so every value of f must be at least 1. And than we can to see that f increasing with increasing x, so range will be all numbers greater than or equal to 1.
Example
Which of the following represents the set of all values of x for which is defined the function:
Answer:
(C )
Both “domain rules” apply:
(x – 6) not equal 0 and (x + 4) ≥ 0. So 
and x ≥ – 4
