determine whether the vector is in the image of A I am having a really hard time trying to understand what "the image of A" means and how to start such a question. 
I was thinking of creating a matrix with b1 as the augumented part of the matrix.
The image of a function $f: X \rightarrow Y$ is the set of elements of $Y$ that are the result of applying $f$ to some $x$ in $X$.

Any $3\times 3$ matrix $A$ gives rise to a linear transformation $T_A$ given by $T_A(\vec{x})=A \vec{x}$.

In other words, a vector $\vec{b}$ is in the image of $A$ if there exists a vector $\vec{x}$ such that $A\vec{x} = \vec{b}$.

Hint: What has this got to do with solving a system of linear equations?
Turn A into the upper echelon matrix U...the columns of A that correspond to the columns of U with the pivots of U are the vectors of the basis for the image of A