Unit 3 - Objective 2 - Domain and Range

The complete set of possible values of the independent variable is called the domain of the function, and the complete set of all possible resulting values of the dependent variable is called the range of the function.  The domain of a function defined by an equation includes all real numbers except:

• any number for which the denominator is zero
• any number for which the expression defined by the equation is not a real number (for example a negative number under a square root)
Examples:
1. y = 2x + 5

It is possible to replace x with any real number and get a value of y.  Since we double every value of x and then add 5 to it, we can do this for any value of x. It is also possible to generate any value for y by selecting appropriate x values.

 Therefore: Domain: all real numbers Range: all real numbers
2. y = x2 - 4

It is possible to replace x with any real number and get a value of y.  However since x2 is never negative, x2 - 4 is never less than - 4 so the range will be all real numbers greater than or equal to - 4.

 Therefore: Domain: all real numbers Range: all real numbers such that y > - 4
3. y =

The square root is real if the quantity under the square root is greater than or equal to zero.
We need x - 5 > 0 or x > 5.

The range cannot be negative so y will be any real number greater than or equal to zero.

 Therefore: Domain: all real numbers such that x > 5 Range: all real numbers such that y > 0
4. y =

The denominator cannot equal zero so if we set the denominator equal to zero we will find the values which are excluded from the domain.

x - 3 = 0
x = 3

So the domain is all real numbers except x = 3.  The range will take on all real numbers except y = 0 because no matter what you plug in for x in the equation, the value of y will never be zero.

 Therefore: Domain: all real numbers, such that x cannot equal 3 Range: all real numbers, such that y cannot equal 0