1. The problem states that we have a function $f : A \to B$ with sets $A = \{1, 2, 3\}$ and $B = \{4, 5, 6\}$. The function values are given as $f(1) = 4$, $f(2) = 5$, and $f(3) = 6$. 2. The image of a set $A$ under a function $f$ is the set of all outputs $f(a)$ for each $a \in A$. 3. Since $f(1) = 4$, $f(2) = 5$, and $f(3) = 6$, the image of $A$ is the set $\{4, 5, 6\}$. 4. Comparing with the options, the correct answer is option a. $\{4, 5, 6\}$. Final answer: $\boxed{\{4, 5, 6\}}$