English: Counter-examples with f:R→R defined by f(x) = x² for function image/preimage laws that might intuitively be expected but do not hold in general. We use the interval names A₁=[-4,+2], A₂=[-2,+4], A₃=[-2,+2], A₄=[0,+2] in f′s domain, and B₁=[0,16], B₂=[0,4], B₃=[-4,+4] in f′s codomain.

• While f(A∩A′) ⊆ f(A) ∩ f(A′) holds generally, f(A∩A′) ⊇ f(A) ∩ f(A′) does not:

e.g. f(A₁∩A₂) = f(A₃) = B₂ ⊅ B₁ = B₁ ∩ B₁ = f(A₁) ∩ f(A₂).

• While f(f⁻¹(B)) ⊆ B holds generally, f(f⁻¹(B)) ⊇ 'B does not:

e.g. f(f⁻¹(B₃)) = f(A₃) = B₂ ⊅ B₃.

• While f⁻¹(f(A)) ⊇ A holds generally, f⁻¹(f(A)) ⊆ A does not:

e.g. f⁻¹(f(A₄)) = f⁻¹(B₂) = A₃ ⊄ A₄.