Describing Functions (Properties I)
We have identified a real valued function y = f(x) of a real variable x with a set of points or ordered pairs
F = graph(f) = { (x,y) | x belongs to domain f and y = f(x)}
in the plane.
Digression: Because F is the graph of a function, for each real number a, the vertical line x = a has at most one point (x,y) of intersection with F. Further if x = a belongs to the domain of f or F, the intersection has at most one point, namely (x,y) = (a, f(a)).
Definitions - More on Injectivity
Suppose y = f(x) denotes or indicates a function f. Then the function f is
- injective if and only if for each element b of the
range, there is at most one element a of the domain such that
f(a) = b.
In other words, the equation f(x) = y has at most one solution x for each element y in the range of f. Or, if f(x) = f(a) for some element a of the domain of f, then x = a.
- one-to-one if and only if f is injective. In other words, saying f is one-to-one has the same meaning as saying f is injective.
- many-to-one on a proper or improper subset S of the domain of f
if and only if f is not one-to-one on S.
- two-to-one if and only if for each element a of its domain such that the equation f(x) = f(a) has exactly two solutions, one of them being a.
- N-to-one if and only if for each element a of S such that the equation f(x) = f(a) has exactly N solutions, one of those solutions being a.
In the definition of N-to-one, N has to be a whole number.
Example of one-to-one. The graph of f(x) = 4 - 2x follows.
Each horizontal line has at most one point of intersection with the graph of y = f(x). Here y = b = 4-2x when and only when b-4 =2x or x =½ (b-4) = a. So for each real number b, there is a unique number a = ½ (b-4) for which f(a) = b.
Related topic: Horizontal Line Rule and Calculation of Inverse functions.
The range of the function y = f(x) = 4 - 2x includes all real numbers b. So
f: IR --> IR
is both surjective and injective - a bijection.
Example of Two to One (not one to one)
The function y = x2 is two to one for all points in its range [0,+oo) = [0.+oo[.
For each real number b in the range [0,+oo) = [0.+oo[ of f(x) = x2, there a two points x = a and x = -a in the domain of f for which f(x) = b.
On its range [0,+oo) = [0.+oo[, the function f(x) = x2, is many to one, more precisely, two to one on its domain (-oo,+oo) = ]-oo,+oo[ = IR
The domain of f is the real line. The range of f is the interval [0, +oo) = [0,+oo[ = the set of non-negative real numbers. The function is a surjection of the real number line onto the set of non-negative real numbers [0, +oo) = [0,+oo[
Example of Many-To-One (Not one to one)
The height y = f(x) of an observed bird varies with it distance from a starting point. At different distances say x = d1 and x= d2 the bird has the same height: That is, f(d1) = f(d2)
Further Definitions - Optional
Suppose y = f(x) denotes or indicates a function f. Then the function f is
- injective on a proper or improper subset S of the domain of f if and only if for each real number b there is at most one element a of S such that f(a) = b. In other words, the equation f(x) = y has at most one solution x.
- one-to-one on proper or improper subset S of the domain of f if and only if f is injective on S. In other words, saying f is one-to-one on S has the same meaning as saying f is injective on S.
- many-to-one on a proper or improper subset S of the domain of f
if and only if f is not one-to-one on S.
- two-to-one on a proper or improper subset S of the domain of f if and only if for each element a of S such that the equation f(x) = f(a) has exactly two solutions, one of which will be x = a.
- N-to-one on a proper or improper subset S of the domain of f if and
only if for each element a of S such that the equation f(x) = f(a) has
exactly N solutions, one of which will be x = a.
In the definition of N-to-one, N has to be a whole number.
Exercise for students & teachers: Give examples to illustrate the further definitions.