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Choice A is correct. The x-coordinate of any y-intercept of a graph is $0$. Substituting $0$ for $x$ in the given equation yields $g\left(0\right)=11{\left(\frac{1}{12}\right)}^0$. Since any nonzero number raised to the $0\text{th}$ power is $1$, this gives $g\left(0\right)=11·1$, or $g\left(0\right)=11$. The y-intercept of the graph is, therefore, the point $\left(0,11\right)$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.