If f(x1) = f (x2) ⇒ x1 = x2 ∀ x1 x2 ∈ A then the function f: A → B is

The correct option is one-one.

Concept and Definition:
A function
$f:A\to B$
is a rule that associates each element in set A (the domain) with an element in set B (the codomain).

A function is defined as a one-one (or injective) function if distinct elements in the domain have distinct images in the codomain. In other words, no two different elements in the domain map to the same element in the codomain.

Mathematically, we can express this condition in two equivalent ways:
1. If two inputs are different, their outputs must be different:
$x_1\ne x_2\Rightarrow f\left(x_1\right)\ne f\left(x_2\right)$
2. The contrapositive statement (if the outputs are equal, the inputs must be equal):
$f\left(x_1\right)=f\left(x_2\right)\Rightarrow x_1=x_2$
for all
$x_1,x_2\in A$
.

Conclusion:
Since the statement in the question,
$f\left(x_1\right)=f\left(x_2\right)\Rightarrow x_1=x_2\forall x_1,x_2\in A$
is the formal mathematical definition of injectivity, the function is classified as a one-one function.