Relationship between a utility function and preference

Question

Consider the following utility functions:

Q1. Explain what the relationship between a utility function and preference is.

Q2. What types of preferences U₁(x₁,x₂), U₂(x₁,x₂), U₄(x₁,x₂) and U₆(x₁,x₂) represent, respectively?

Q3. For each of the utility functions U₂(x₁,x₂), U₃ (x₁,x₂), and U₄(x₁,x₂), show whether it has the same preferences as U₁(x₁,x₂)

Q4. Suppose U₅(x₁,x₂) and U₆(x₁,x₂) represent the preferences for consumer E and consumer F, respectively. Determine whether each commodity x₁ and x₂ is an economic good, an economic bad or an economic neutral for E and F.

Q5. Do U₅(x₁,x₂) and U₆(x₁,x₂) represent the same preference? Explain.

Q6. For each of U₃ (x₁,x₂), U₄(x₁,x₂), U₅(x₁,x₂) and U₆(x₁,x₂) check whether they represent monotonic preferences? Explain in detail by marginal utilities.