Answer
$f(x)=\frac{4}{3}\pi (3r^{2}+3r+1)$
Work Step by Step
Lets calculate the amount of air needed to inflate a balloon with radius (r+1)
and subtract the air needed for a balloon with radius r.
$f(x)=V(r+1)-V(r)=\frac{4}{3}\pi (r+1)^{3}-\frac{4}{3}\pi r^{3}=\frac{4}{3}\pi ((r+1)^{3}-r^{3})$
*$(r+1)^{3}-r^{3}= r^{3}+3r^{2}+3r+1-r^{3}=3r^{2}+3r+1$
So,
$f(x)=V(r+1)-V(r)=\frac{4}{3}\pi (r+1)^{3}-\frac{4}{3}\pi r^{3}=\frac{4}{3}\pi ((r+1)^{3}-r^{3})=\frac{4}{3}\pi (3r^{2}+3r+1)$
