The graphs of f(x) = p^x + q and a straight line g are drawn below. A(3; 4) is a point of intersection of f and g. B (0;-3) and C are the y- and x-intercepts of f respectively. E and D are the x- and y-intercepts of g respectively.

The function $f(x) = p^x + q$ has a horizontal asymptote at $y=q$. From the graph, the dashed line indicates the horizontal asymptote is $y=-4$.
Therefore, $q = -4$.

The point B $(0;-3)$ is the $y$-intercept of $f(x)$. Substitute $x=0$ and $f(x)=-3$ into the equation $f(x) = p^x + q$.

Since $p^0 = 1$ (for $p \neq 0$), and we found $q=-4$:

This confirms that $q=-4$ is consistent with the $y$-intercept.

The point A $(3;4)$ lies on the graph of $f(x)$. Substitute $x=3$, $f(x)=4$, and $q=-4$ into the equation $f(x) = p^x + q$.

Add 4 to both sides of the equation:

Take the cube root of both sides:

The values are $p=2$ and $q=-4$.
$\boxed{p=2, q=-4}$