Finding value of $c$ such that the range of the rational function $f(x) = \frac{x^2 + x + c}{x^2 + 2x + c}$ does not contain $[-1, -\frac{1}{3}]$

Question

I am comfortable when I am asked to calculate the range of a rational function, but how do we do the reverse? I came across this problem.

If $$f(x)= \frac{x^2 + x + c}{x^2 + 2x + c}$$ then find the value of $c$ for which the range of $f(x)$ does not contain $[-1, -\frac{1}{3}]$.