Discrete Mathematics Demystified

CHAPTER 4 Functions and Relations 63
Let f : {x ∈ R : x ≥ 0} →R be given by f (x) = x 3 . Then f is one-to-one but
f is not onto. There certainly is a function g : R →{x ∈ R : x ≥ 0} such that
(g ◦ f )(x) = x for all x ∈{x ∈ R : x ≥ 0} [namely g(x) = 3√ |x|]. But there is no
function h : R →{x ∈ R : x ≥ 0} such that ( f ◦ h)(x) = x for all x.
We have established that if f : S → T has an inverse then f must be one-toone
and onto. The converse is true too, and we leave the details for you to verify.
A function f : S → T that is one-to-one and onto (and therefore invertible) is
sometimes called a set-theoretic isomorphism or a bijection. It is also common to
use the terminology one-to-one correspondence.
EXAMPLE 4.19
The function f : R → R that is given by f (x) = x 3 is a bijection. You should
check the details of this assertion for yourself. The inverse of this function f is the
function g : R → R given by g(x) = x 1/3 .
We leave it as an exercise for you to verify that the composition of two bijections
(when the composition makes sense) is a bijection.
4.5 Types of Functions
The most elementary and easiest understood type of function is the polynomial
function. A polynomial has the form
p(x) = a0 + a1x + a2x 2 + a3x 3 +···+akx k
We call a0 the constant coefficient, a1 the linear coefficient, a2 the quadratic coefficient,
a j the jth-degree coefficient, and ak the top-order coefficient.
A polynomial is easy to understand because we can calculate its value at any
point x by simply plugging in x and then multiplying and adding. A computer can
calculate values of a polynomial very rapidly and easily.
Another important type of function is the exponential function. For example,
g(x) = 2 x is an exponential function. This function is easy to calculate when x
is an integer; for example, f (4) = 2 4 = 2 · 2 · 2 · 2 = 16. It is more difficult to
calculate when x is a noninteger; some approximation procedure would probably
be needed (this is how your calculator effects the computation).
You probably know that, when you put money in the bank, and it earns interest,
then it grows according to a rule given by an exponential function. For example,
suppose that you put $1000 in the bank and it earns interest annually at the rate of
5%. Then we have