All Algebra II Resources
Example Questions
Example Question #21 : Range And Domain
Find the domain and range of the function . Express the domain and range in interval notation.
Domain
Range
(all real numbers)
Domain
Range
Domain
Range
Domain
(all real numbers)
Range
Domain
Range
Domain
Range
Finding the Domain
The domain of a function is defined as the set of all valid input values of overwhich the function is defined. The simple rule of thumb for rational functions is that all real numbers will work except for those in which denominator is zero since division by zero is not allowed.
Set the denominator to zero and solve for ,
The function is therefore defined everywhere except at . Therefore the domain expressed in interval notation is,
Note that the open parentheses indicate that is not in the domain, but may become arbitrarily close to .
Finding the Range
The range of a function is defined as the set of all outputs spanning the domain. Finding the range can be achieved by finding the domain of the inverse function. First solve for to obtain the inverse function,
Multiply both sides by ,
Distribute ,
Move all terms with to one side of the equation,
Factor and solve for
The inverse function is therefore,
Find the domain of the inverse function,
The range of is the domain of , which is:
If you look at the plots for the function (in blue) and (in red and labeled as in the figure) you can see the asymptotic behavior of as approaches and of as approaches .