Question

We say a function f: A -> B is injective (one-to-one) iff for all xE A, r2A, f(x1) = f(x2) implies r = r2. What does it mean for a function not to be injective?

Answer 1

By definition a function
`f:A\rightarrow B` is injective if and only if
`f(x_1)=f(x_2)` implies
`x_1=x_2`. Notice that this means that different elements of the domain have different images. That is why injective functions are also called one-to-one, because each element from A has only one image in B.

Now, if a function is not injective means that there are, at least, two elements of the domain with the same image. A very good example is the function
`f(x)=x^2`. Recall that
`f(-1) = (-1)^2=(1)^2=f(1)`.

For real functions of a real variable
`f:\mathbb{R}\rightarrow\mathbb{R}` we have a geometrical interpretation of injective property. The function
`f` is injective if for each line we draw parallel to the X-axis, it has, at most, one intersect with the graph of
`f`. Then, if one line has more than one intercept, then the function is not injective.