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