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

Answer:

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.