Question 1129635
An example of a function, f(x), with a domain of ({{{0}}},{{{5}}}] and a range of [{{{0}}},{{{infinity}}}) is following:

 {{{f(x) = (5 - x)/(2x)}}} when {{{0 < x <=5}}}

 to prove:
{{{x=0}}} , then {{{f(x)= (5-0)/(0) = indefinite =infinity}}}

{{{x = 5}}}, then {{{f(x)= 0/10= 0}}}

thus, function of  {{{f(x) = (5 - x)/(2x)}}} have a domain of ({{{0}}},{{{5}}}] and a range of [{{{0}}},{{{infinity}}}) 



another example:

The domain is ({{{0}}},{{{5}}}], which includes all real numbers between {{{0}}} and {{{5}}}, excluding {{{0}}} and including {{{5}}}. 

The functions that do not have {{{0}}} in its domain are for example: {{{y=1/x}}} . 

The domain of {{{y=1/x }}}is all the real numbers except {{{0}}},i.e., ({{{-infinity}}},{{{0}}}) U ({{{0}}},{{{infinity}}}). 

To find a way to {{{exclude}}} the negative numbers from the domain is by remembering that the {{{radicand}}} of a square root {{{cannot}}} be {{{negative}}}.

 So having that in mind, and applying that to the example above gives the following result: {{{y=1/sqrt(x)}}}, and its domain is ({{{0}}},{{{infinity}}}). 


The domain also {{{cannot }}}include {{{any}}}{{{ number}}}{{{ greater}}} than {{{5}}}, therefore {{{y=sqrt(5-x) / sqrt(x)}}} 

This function requires {{{5-x >= 0}}} (for the top) and {{{x> 0}}} (for the bottom). 
So the domain of {{{y=sqrt(5-x) / sqrt(x)}}}  is ({{{0}}},{{{5}}}]. 

Now about the {{{range}}}: square roots are {{{non-negative}}}, so {{{y}}} {{{cannot}}}{{{ be}}} {{{negative}}}. 

When {{{x=5}}}, {{{y=0}}}, which is the smallest value of {{{y}}}.
 
On can note that {{{x}}} gets closer and closer to {{{0}}}, the numerator gets closer and closer to {{{sqrt(5)}}} , and the {{{denominator}}} gets even to {{{0 }}}(still being {{{positive}}}). 

Since {{{y= sqrt(5-x)/sqrt(x) = sqrt((5-x)/x )}}}

the square root function is an increasing function, we can just consider{{{ (5-x)/x}}} 

As the denominator gets smaller, the expression {{{(5-x)/x}}} gets bigger. 

So, the function {{{y= sqrt((5-x)/x )}}} satisfies the requirement of the posted question.