Why Read This Book? - Index of

126 Chapter 4 Functions
Example 4.2.2 Show that f : R → R defined by f(x) = x 3 − 1 is a surjection.
Solution You may recall from pre-calculus that the graph of a polynomial function
of odd degree extends all the way up and down the y-axis, so that any value
chosen on the y-axis has a pre-image on the x-axis. But how do we prove f is
onto? We have to choose an arbitrary y in the codomain and work backwards to
find some x in the domain that maps to it. In other words, we must find a value of
x that makes the equation y = f(x) true. Here is the proof.
Pick any real number y, and let x = 3√ y + 1. By Theorem 3.9.1, x is a real
number, and
Thus f is onto. �
f(x) = f( 3� y + 1) =[ 3� y + 1] 3 − 1 = y + 1 − 1 = y (4.4)
EXERCISE 4.2.3 Show that the function T : Nn → Nm n from Example 4.1.5 is a
surjection.
EXERCISE 4.2.4 Let f :[0, 2] →[1, 7] be defined by f(x) = 3x + 1. Show that
f is a surjection.
EXERCISE 4.2.5 Let A be a non-empty set, and i : A → A the identity function
(Exercise 4.1.8). Show that i is onto.
EXERCISE 4.2.6 What does it mean to say that a function is not onto?
EXERCISE 4.2.7 Show that f :[−2, 3] →[0, 10] defined by f(x) = x 2 + 1is
not onto.
Definition 4.2.8 Suppose f : A → B is a function. Then f is said to be one-toone
provided f(x1) = f(x2) implies x1 = x2 for all x1,x2 ∈ A. That is,
f is one-to-one ⇔ (∀x1,x2 ∈ A)([f(x1) = f(x2)] →[x1 = x2]) (4.5)
As a notational shorthand, we write f : A 1-1
−→ B. A one-to-one function is also
called an injection.
Note that the definition of one-to-one in (4.5) is the converse of property F2,
which says
(∀x1,x2 ∈ A)([x1 = x2] →[f(x1) = f(x2)])