408
Relations, Functions, and Matrices
45. Let S = 51, 2, 3, 46, T = 51, 2, 3, 4, 5, 66, and U = 56, 7, 8, 9, 106. Also, let
f = 5(1, 2), (2, 4), (3, 3), (4, 6)6 be a function from S to T, and let
g = 5(1, 7), (2, 6), (3, 9), (4, 7), (5, 8), (6, 9)6 be a function from T to U. Write the ordered pairs in the
function g + f .
46. a. Let f : R S Z be defined by f (x) = :x;. Let g: Z S N be defined by g(x) = x
2
. What is (g + f )(−4.7)?
b. Let f map the set of books into the integers where f assigns to each book the number of words in its title.
Let g: Z S Z be given by g(x) = 2x. What is (g + f ) (this book)?
c. Let f map strings of alphabetical characters and blank spaces into strings of alphabetical consonants
where f takes any string and removes all vowels and all blanks. Let g map strings of alphabetical
consonants into integers where g maps a string into the number of characters it contains. What is (g + f )
(abraham lincoln)?
47. Let f : N S N be defined by f (x) = x + 1. Let g: N S N be defined by g(x) = 3x. Calculate the value
of the following expressions.
a. (g + f )(5)
b. ( f + g)(5)
c. (g + f )(x)
d. ( f + g)(x)
e. ( f + f )(x)
f. ( g + g)(x)
48. The following functions map R to R. Give an equation describing the composition functions g + f and
f + g in each case.
a. f (x) = 6x
3
, g(x) = 2x
b. f (x) = (x − 1)∙2, g(x) = 4x
2
c. f (x) =
49. Let f : S S T and g: T S U be functions.
a. Prove that if g + f is one-to-one, so is f.
b. Prove that if g + f is onto, so is g.
c. Find an example where g + f is one-to-one but g is not one-to-one.
d. Find an example where g + f is onto but f is not onto.
50. a. Let f be a function, f : S S T . If there exists a function g: T S S such that g + f = i S , then g is called a
left inverse of f. Show that f has a left inverse if and only if f is one-to-one.
b. Let f be a function, f : S S T . If there exists a function g: T S S such that f + g = i T , then g is called
a right inverse of f. Show that f has a right inverse if and only if f is onto.
c. Let f: N S N be given by f (x) = 3x. Then f is one-to-one. Find two different left inverse functions
for f.
d. Let f: N
+ S N
+
be given by f (x) = l
x
2
m . Then f is onto. Find two different right inverse functions
for f.
51. For each of the following bijections f: R S R, find f
−1
.
a. f (x) = 2x
b. f (x) = x
3
c. f (x) = (x + 4)∙3
52. Let f and g be bijections, f : S S T and g: T S U . Then f
−1
and g
−1
exist. Also, g + f is a bijection from
S to U. Show that (g + f )
−1
= f
−1 + g
−1
.
Relations, Functions, and Matrices
45. Let S = 51, 2, 3, 46, T = 51, 2, 3, 4, 5, 66, and U = 56, 7, 8, 9, 106. Also, let
f = 5(1, 2), (2, 4), (3, 3), (4, 6)6 be a function from S to T, and let
g = 5(1, 7), (2, 6), (3, 9), (4, 7), (5, 8), (6, 9)6 be a function from T to U. Write the ordered pairs in the
function g + f .
46. a. Let f : R S Z be defined by f (x) = :x;. Let g: Z S N be defined by g(x) = x
2
. What is (g + f )(−4.7)?
b. Let f map the set of books into the integers where f assigns to each book the number of words in its title.
Let g: Z S Z be given by g(x) = 2x. What is (g + f ) (this book)?
c. Let f map strings of alphabetical characters and blank spaces into strings of alphabetical consonants
where f takes any string and removes all vowels and all blanks. Let g map strings of alphabetical
consonants into integers where g maps a string into the number of characters it contains. What is (g + f )
(abraham lincoln)?
47. Let f : N S N be defined by f (x) = x + 1. Let g: N S N be defined by g(x) = 3x. Calculate the value
of the following expressions.
a. (g + f )(5)
b. ( f + g)(5)
c. (g + f )(x)
d. ( f + g)(x)
e. ( f + f )(x)
f. ( g + g)(x)
48. The following functions map R to R. Give an equation describing the composition functions g + f and
f + g in each case.
a. f (x) = 6x
3
, g(x) = 2x
b. f (x) = (x − 1)∙2, g(x) = 4x
2
c. f (x) =
a. Prove that if g + f is one-to-one, so is f.
b. Prove that if g + f is onto, so is g.
c. Find an example where g + f is one-to-one but g is not one-to-one.
d. Find an example where g + f is onto but f is not onto.
50. a. Let f be a function, f : S S T . If there exists a function g: T S S such that g + f = i S , then g is called a
left inverse of f. Show that f has a left inverse if and only if f is one-to-one.
b. Let f be a function, f : S S T . If there exists a function g: T S S such that f + g = i T , then g is called
a right inverse of f. Show that f has a right inverse if and only if f is onto.
c. Let f: N S N be given by f (x) = 3x. Then f is one-to-one. Find two different left inverse functions
for f.
d. Let f: N
+ S N
+
be given by f (x) = l
x
2
m . Then f is onto. Find two different right inverse functions
for f.
51. For each of the following bijections f: R S R, find f
−1
.
a. f (x) = 2x
b. f (x) = x
3
c. f (x) = (x + 4)∙3
52. Let f and g be bijections, f : S S T and g: T S U . Then f
−1
and g
−1
exist. Also, g + f is a bijection from
S to U. Show that (g + f )
−1
= f
−1 + g
−1
.