Question 1129635: Give an example of a function, f(x), with a domain of (0,5] and a range of [0,∞)
Answer by MathLover1(20850)   (Show Source):
You can put this solution on YOUR website! An example of a function, f(x), with a domain of ( , ] and a range of [, ) is following:
 when 
to prove:
 , then 
 , then 
thus, function of have a domain of (,] and a range of [,)
another example:
The domain is (,], which includes all real numbers between and , excluding and including .
The functions that do not have in its domain are for example:  .
The domain of  is all the real numbers except ,i.e., ( ,) U (,).
To find a way to  the negative numbers from the domain is by remembering that the  of a square root  be  .
So having that in mind, and applying that to the example above gives the following result:  , and its domain is (,).
The domain also  include    than , therefore 
This function requires  (for the top) and  (for the bottom).
So the domain of is (,].
Now about the  : square roots are  , so   .
When  ,  , which is the smallest value of .
On can note that  gets closer and closer to , the numerator gets closer and closer to  , and the  gets even to  (still being  ).
Since 
the square root function is an increasing function, we can just consider
As the denominator gets smaller, the expression  gets bigger.
So, the function  satisfies the requirement of the posted question.
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