Definition of Domain and Range of a function

Domain and Range of a Function

Definitions of Domain and Range

Domain

It is the complete set of possible values of the independent variable.

Technical Definition:

The domain is the set of all possible x-values which will make the function "work", and will output real y-values.

When We find the domain, remember:

• The denominator (bottom) of a fraction cannot be zero
• The number under a square root sign must be positive in this section

How to find the domain

In general, we determine the domain of each function by looking for those values of the independent variable (usually x) which we are allowed to use. (Usually we have to avoid 0 on the bottom of a fraction, or negative values under the square root sign).

Range

It is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain.

Technical Definition:

It is the resulting y-values we get after substituting all the possible x-values.

How to find the range

• The range of a function is the spread of possible y-values on Y-axis (minimum y-value to maximum y-value)
• Substitute different x-values into the expression for y to see what is happening.

     *     Make sure you are looking for minimum and maximum values of y.

     *     Draw a sketch! In mathematics, it's very true that a picture is            worth thousand words.

Example 2

The graph of the curve y = sin x shows the range in  betweeen −1 and 1.

The domain of y = sin x  “can be any real number” because if you put any real number at the place of x you will get y in between -1 to 1.

From the calculator experiment, we can see the range  y is −1 ≤ y ≤ 1.

                                                                                                                                                                                                                                                                                                                      Nitesh