Algebra II : Domain and Range

Study concepts, example questions & explanations for Algebra II

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Example Questions

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Example Question #21 : Properties Of Functions And Graphs

Find the domain and range of the function . Express the domain and range in interval notation.

 

Possible Answers:

Domain

Range

Domain

Range

 

Domain

 

Range

  (all real numbers) 

Domain

 (all real numbers) 

Range

Domain

 

Range 

Correct answer:

Domain

 

Range 

Explanation:

 

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 .

 

Problem 1 plot2

 

 

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