epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

66 The Nuts and Bolts of Proof, Third Edition
3. The equation x^ — b = 0, with b real number, has a unique solution.
4. There exists a unique second degree polynomial P such that P(0) = — 1,
P(l) = 3, andP(-l)-:2.
5. The graphs of the functions f(x) = x^ and g(x) = —x^ — 2x have a
unique intersection point.
Outline the proofs of the following statements, filling in all details as needed:
1. Let/be a function defined for all real numbers. If/is one-to-one and
onto, its inverse exists and it is unique. (See the front material of the
book for the definition of one-to-one and onto.)
Proof: We need to find a function g defined for all real numbers such
that:
9 o/(x) = g(fix)) = X
fogiy)=f(9(y))
for all real numbers x and y, where x indicates a number in the domain
of/ and y indicates a number in the range of/
The function g is defined as follows. Let y be any real number,
then g{y) = x, where x is the unique number with the property that
fix) = y.
(How do we know that such a number x exists and it is unique for any
value of yl)
We claim that the function g defined in this way is indeed the inverse of
the function /
Check: Let XQ be a real number such that/(xo) = yo. Thus,
= y
g of{xo) = g(f(xo)) = g(yo) = XQ.
On the other hand, if g(yo) = XQ, it follows that f(xo) = yo- So,
/ o d(yo) =f{g(yo)) =f(xo) = yo-
Assume that there is another function h that is the inverse of /
Therefore,
h of(x) = X
f o h(y) = y
for all real numbers x and y. Thus,
foh{y) =
y=fog(y)