Question 990326
I assume that you meant {{{g(x)=5/(x+9)}}}= 5/(x+9) ,
because {{{g(x)=5/x+9}}}= 5/x+9 is undefined for {{{x=0}}} .


The average rate of change of the function between {{{x=0}}} to {{{x=h}}} is
{{{(g(h)-g(0))/(h-0)}}} ,
the change in the value of the function,
divided by the corresponding change of the variable.
The values of the function are:
for {{{x=0}}} , {{{g(0)=5/(0+9)=5/9}}} ,
for {{{x=h}}} , {{{g(h)=5/(h+9)}}} .
As the value of {{{x}}} changes by {{{h=h-0}}} units from {{{x=0}}} to {{{x=h}}} ,
the value of {{{g(x)}}} changes by
{{{g(h)-g(0)=5/(h+9)-5/9=5*9/(9(h+9))-5(h+9)/(9(h+9))=45/(9(h+9))-(5h+45)/(9(h+9))=(45-5h-45)/(9(h+9))=-5h/(9(h+9))}}} .
The average rate of change of the function between {{{x=0}}} to {{{x=h}}} is
{{{((-5h/(9(h+9))))/h=-5/(9(h+9))}}}