First factor the denominator:

Because

rewrite the right-hand side as a single fraction:

To match the denominator

multiply numerator and denominator by $x+3$:

Since the two rational expressions are equal for all allowed $x$, their numerators must be equal:

Expand the right side:

Now match coefficients with

This gives

and

so

and then

.
Therefore,

Because the graph passes through $(0,3)$, substitute $x=0$ and $f(0)=3$ into

This gives

so $b=-6$. Next, use $(4,7)$:

Substituting $b=-6$ gives

so $4a-6=14$, which means $4a=20$ and $a=5$. For a rational function of the form

the degrees of the numerator and denominator are the same, so the horizontal asymptote is the ratio of the leading coefficients:

Therefore, the equation of the horizontal asymptote is

Set up the condition that the cost per student is at most $36$:

Subtract $12$ from both sides:

Because $x>0$, multiply both sides by $x$ without changing the inequality:

Now divide by $24$:

So the least possible whole number of students is $100$.

Since $f(x)=\frac{1}{x+b}$, use any row of the table to find $b$. From $f(2)=\frac{1}{4}$, we have

so $2+b=4$, which gives $b=2$. This matches the other rows because $f(5)=\frac{1}{5+2}=\frac{1}{7}$ and $f(8)=\frac{1}{8+2}=\frac{1}{10}$. Then

Factor the numerator: $x^2-16=(x-4)(x+4)$. Then

because $x\ne 4$, so dividing by $x-4$ is allowed. The equivalent expression is $x+4$.