Question

To keep her dog from running away while she talks to a friend, susan pulls gently on the dog's leash with a constant force given by f⃗ = (2.3 n )x^ + (1.2 n )y^. part a how much work does she do on the dog if its displacement is d⃗ = (0.34 m )x^,

Answer 1

A force is doing work if there is a displacement of the object, on which the force is acting, along with the direction of the force.
Mathemactly the work is defined as a dot product between the force and the displacement vector.
$W=\textbf{F} \cdot \textbf{L}=\sum_{i} F_iL_i$
Where $F_i and L_i$ are the components of these vectors.
In our case we have the displacement only in the x-direction:
$W=F_xL_x=2.3\cdot0.34=0.782 N$