Answer
The vector field $F$ is irrotational.
Work Step by Step
The vector field $F$ will be irrotational when $curl F=0$
consider $F=A i+B j+C k$
$curl F=[C_y-B_z]i+[A_z-C_z]j+[B_x-A_y]k$
Given: $F(x,y,z)=f(x) i+g(y) j+h(z) k$
Here, we have $curl F= \nabla \times F=(\dfrac{\partial h(z)}{\partial y}-\dfrac{\partial g(y)}{\partial z})i+(\dfrac{\partial f(x)}{\partial z}-\dfrac{\partial h(z)}{\partial x})j+(\dfrac{\partial g(y)}{\partial x}-\dfrac{\partial f(x)}{\partial y})k=0$
Hence, we can see that the vector field $F$ is irrotational.
