278 CHAPTER 4 Functions
Which of the sets F 1 , F 2 , and F 3 are onto functions?
(a) Fl, F 2 , F 3
(b) F 2 , F 3
(c) F 1 , F 3
(d) F 1 , F 2- Which of the following is the correct definition of a decreasing function?
(a) If for all x 1 , x 2 E X, x1 > x2 implies F(xl) < F(x 2 ).
(b) If for all Xl, x2 E X, xI < X2 implies F(x1) < F(x 2 ).(c) If for all xl, x 2 E X, x 1 < x2 implies F(xl) < F(x 2 ).
(d) None of the above.5. For n = 0, 1 .... 10, list the values of the following functions defined recursively:
(a) function s(n):
if n = 0 then
return (0)
else
return (2n + s(n - 1))
(b) function p(n):
if n = 0 then
return (1)
else
return (n .p(n - 1))
(c) function d(n):
if n = 10 then
return (100)
else
return (d(n + 1) - 10)- Express the function F(x) = x^2 + 3x + 2 as the composition of two simpler func-
tions, neither of which is the identity function.
7. Let F and G be the functions defined on {1, 2, 3, 4, 5) as follows:
F: 1 - 2 G: 1 - 3
2-- 3 2-+ 1
3-- 5 3-- 2
4 4 4 55-* 1 5 * 4Prove that F and G are 1-1. Also, prove that F o G and G o F are both 1-1 and onto.
8. Let X = 1-1, 0, 1, 2) and Y = {-4, -2, 0, 21. Define the function F : X -- Y as
F(x) = x2 - x. Prove F is neither 1-1 nor onto.