s
Section 5.4 Functions
405
15. Which of the following definitions describe functions from the domain to the codomain given? Which
functions are one-to-one? Which functions are onto? Describe the inverse function for any bijective
function.
a. f: Z S N where f is defined by f (x) = x
2
+ 1
b. g: N S Q where g is defined by g(x) = 1∙x
c. h: Z × N S Q where h is defined by h(z, n) = z∙(n + 1)
d. f: 51, 2, 36 S 5 p, q, r6 where f = 5(1, q), (2, r), (3, p)6
e. g: N S N where g is given by g(x) = 2
x
f. h: R
2 S R
2
where h is defined by h(x, y) = (y + 1, x + 1)
16. Which of the following definitions describe functions from the domain to the codomain given? Which
functions are one-to-one? Which functions are onto? Describe the inverse function for any bijective
function.
a. f: Z
2 S N where f is defined by f (x, y) = x
2
+ 2y
2
b. f: N S N where f is defined by f (x) = e
x∙2 if x is even
x + 1 if x is odd
c. g: R S R where g is defined by g(x) = 1∙ "(x + 1)
d. f: N S N where f is defined by f (x) = e
x + 1 if x is even
x − 1 if x is odd
e h: N
3 S N where h is given by h(x, y, z) = x + y − z
f. g: N
2 S N
3
where g is defined by g(x, y) = (y, x, 0)
17. Let f: R S R be defined by f (x) = x
n
, where n is a fixed, positive integer. For what values of n is f
bijective?
18. Let f: R S R be defined by f (x) = n2
x
, where n is a fixed, positive integer. For what values of n is f an
onto function?
19. Let A = 5x, y6 and let A* be the set of all strings of finite length made up of symbols from A. A function
f : A* S Z is defined as follows: For s in A*, f(s) = the length of s. Is f one-to-one? Prove or disprove.
Is f onto? Prove or disprove.
20. Let A = 5x, y6 and let A* be the set of all strings of finite length made up of symbols from A. A function
f : A* S Z is defined as follows: For s in A*, f(s) = the number of x’s minus the number of y’s. Is f oneto-one? Prove or disprove. Is f onto? Prove or disprove.
21. Let A = 5x, y6 and let A* be the set of all strings of finite length made up of symbols from A. A function
f : A* S A* is defined as follows: For s in A*, f(s) is the string obtained by writing the characters of s in
reverse order. Is f one-to-one? Prove or disprove. Is f onto? Prove or disprove.
22. Let A = 5x, y6 and let A* be the set of all strings of finite length made up of symbols from A. A function
f : A* S A* is defined as follows: For s in A*, f(s) = xs (the single-character string x followed by s). Is f
one-to-one? Prove or disprove. Is f onto? Prove or disprove.
23. Let P be the power set of 5a, b, c6. A function f: P S Z is defined as follows: For A in P,
f(A) = the number of elements in A. Is f one-to-one? Prove or disprove. Is f onto? Prove or disprove.
24. Let P be the power set of 5a, b6 and let S be the set of all binary strings of length 2. A function f: P S S
is defined as follows: For A in P, f (A) has a 1 in the high-order bit position (left end of string) if and
only if a is in A. f (A) has a 1 in the low-order bit position (right end of string) if and only if b is in A. Is f
one-to-one? Prove or disprove. Is f onto? Prove or disprove.
25. Let S = 5x 0 x [ R and x ≥ 16, and T = 5x 0 x [ R and 0 < x ≤ 16. Find a function f: S S T that is a
bijection.