2.1 • FUNCTIONS AND LINEAR MAPPINGS 53y vw = f(Z) = 11 + iv
Figure 2.2 f maps A onto B; f maps A into R.You will learn in Chapter 5 that if w = f (z) = ez, the inverse image of the point
0 is the empty set- there is no complex number z such that ez = 0.
The inverse image of a set of points, S, is the collection of all points in the
domain that map into S. If f maps D onto R, it is possible for the inverse image
of R to be function as well, but the original function must have a special property:
A function f is said to be one-to-one if it maps distinct points z 1 :f: z2 onto
distinct points f (z1) :f: f (z2). Many times an easy way to prove that a function
f is one-tcrone is to suppose f (zi) = f (z2), and from this assumption deduce
that z 1 must equal z2. Thus, f (z) = iz is one-to-one because if f (zi) = f (z2),
then iz 1 = iz2. Dividing both sides of the last equation by i gives z1 = Z2· Figure
2.3 illustrates the idea of a one-to-one function: Distinct points get mapped to
distinct points.The function f (z) = z^2 is not one-to-one because - i :f: i, but f (i} =
f ( -i) = - 1. Figure 2.4 depicts this situation: At least two different points get
mapped to the same point.
In the exercises we ask you to demonstrate that one-to-one functions give
rise to inverses that are functions. Loosely spe!\king, if w = f ( z) maps the set
A one-to-one and onto the set B , then for each win B there exists exactly one
point z in A such that w = f (z). For any such value of z we can take they f v- ~·
Figure 2.3 A one-to-one function.