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?

Accepted Solution

Answer:By definition a function [tex]f:A\rightarrow B[/tex] is injective if and only if [tex]f(x_1)=f(x_2)[/tex] implies [tex]x_1=x_2[/tex]. 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 [tex]f(x)=x^2[/tex]. Recall that [tex]f(-1) = (-1)^2=(1)^2=f(1)[/tex].For real functions of a real variable [tex]f:\mathbb{R}\rightarrow\mathbb{R}[/tex] we have a geometrical interpretation of injective property. The function [tex]f[/tex] is injective if for each line we draw parallel to the X-axis, it has, at most, one intersect with the graph of [tex]f[/tex]. Then, if one line has more than one intercept, then the function is not injective.