Let $f(a) = \frac{13+a}{3a+7}$ where $a$ is restricted to positive integers. What is the maximum value of $f(a)$? Let $f(a) = \frac{13+a}{3a+7}$ where $a$ is restricted to positive integers. What is the maximum value of $f(a)$?
I tried graphing but it didn't help.  Could anyone answer?  Thanks!

Expanding on @StevenGregory's answer, which is more than satisfactory:
Since $f(a)=\frac{1}{3}+\frac{32}{a+7/3}$, we can see that it's got the same graph as that of $y=\frac{1}{a}$ but:


*

*Translated left by $7/3$

*Then stretched vertically by $32$

*Then translated vertically by $1/3$


These transformations don't change the fact that the right branch is decreasing except that the right branch now starts at $-\frac{7}{3}$, instead of $0$.
