(a) Define

F:z->Z

by the rule

F(n)=2-3n

, for each integer

n

.\ (i) Is

F

one-to-one?\ Suppose

n_(1)

and

n_(2)

are any integers, such that

F(n_(1))=F(n_(2))

. Substituting from the definition of

F

gives that

2-3n_(1)=

\ . Solving this equation for

n_(1)

and simplifying the result gives that

n_(1)=

. Therefore,

F

is\ (ii) Show that

F

is not onto.\ Counterexample:\ Let

m=

. For this value of

m

, the only number

n

with the property that

F(n)=m

is not an integer. Thus,

F

is not onto.\ (b) Define

G:R->R

by the rule

G(x)=2-3x

for each real number

x

. Is

G

onto?\ Scratch work: Let

y

be any real number.\ On a separate plece of paper, solve the equation

y=2-3x

for

x

. Enter the result-an expression in

y

-in the box below.\

x=

\ (1) Is

x

a real number?\ Sums, products, and differences of real numbers are , and quotients of real numbers with nonzero denominators are always real numbers

. Therefore,

x

is a real number\ (2) Does

G(x)=y

?\ According to the formula that defines

G

, when

G

is applied to

x,x

is multiplied by 3 and the result is subtracted from 2 .\ When the expression for

x

that you found above is multiplied by 3 , the result is\ . And when the result is subtracted from 2, you obtain . Thus,\ Hence,\ a number

x

such that

x

is a real number and

G(x)=y

. Therefore,

(a) Define F:zz by the rule F(n)=23n, for each integer n. (i) Is F one-to-one? Suppose n1 and n2 are any integers, such that F(n1)=F(n2). Substituting from the definition of F gives that 23n1= . Solving this equation for n1 and simplifying the result gives that n1= . Therefore, F is (ii) Show that F is not onto. Counterexample: Let m= . For this value of m, the only number n with the property that F(n)=m is not an integer. Thus, F is not onto. (b) Define G:RR by the rule G(x)=23x for each real number x. Is G onto? Scratch work: Let y be any real number. On a separate plece of paper, solve the equation y=23x for x. Enter the result-an expression in y-in the box below. x=3(2y) (1) Is x a real number? Sims. products, and differences of real numbers are , and quotlents of real numbers with nonzero denominators are . Therefore, x (2) Does G(x)=y ? According to the formula that defines G, when G is applied to x,x Is multiplied by 3 and the result is subtracted from 2 . When the expression for x that you found above is multiplied by 3 , the result is . And when the result is subtracted from 2, you obtain Thus, Hence, a number x such that x is a real number and G(x)=y. Therefore