Exercises 24120. Prove that the function F : Z -* Z defined as F(n) = n + 6 is a bijection.
- For each of the following functions, prove that the function is 1-1 or find an appropri-
ate pair of points to show that the function is not 1-1:
(a) F 2 Z- Z
F I)=-n2 F) n^2 for for n n > <^00(b) F :R JRR-
F(x)= x +l forxEQ
12x for xQ(c) F :R -R J
\ +3x+2 forxEQ
F(x)= x3 forxgQ
(d) F Z -Z 2
Fln n I n3 +1 fornoddfor n even
- (a) Find functions from R to R that are:
i. strictly decreasing
ii. decreasing but not strictly decreasing
iii. neither increasing nor decreasing
iv. both increasing and decreasing
(b) Show that no F : -+ R is both increasing and strictly decreasing.
(c) Find a subset X C JR and a function F : X -+ X where F is both strictly increas-
ing and strictly decreasing.- Construct functions with the following properties:
(a) F : N -+ N such that range(F) = N and, for each n E N, there exist exactly two
solutions for the equation F(x) = n.(b) F : N -> N such that, for each n E N, there are exactly n solutions for the equation
F(x) = n.