Exercises 1
7
1.11. Functions . A function f : A » B take s a n inpu t x fro m a domai n
A an d produce s a n outpu t f(x) i n som e rang e B. Fo r mos t o f thi s boo k
we'll assum e tha t A c R
n
an d B c R . Th e se t o f al l output s
f(A) = {f(x):xeA}cB
is called the image o f /. I f the image is the whole range, then / i s called onto
or surjective. I f / map s distinc t point s t o distinc t values , the n / i s calle d
infective o r one-to-one (1-1)I . f / i s bot h injectiv e an d surjective , the n /
is calle d bijective o r a 1-1 correspondence an d / ha s a n invers e functio n
f'l:B^A.
If X i s a subse t o f A, the n f(X) i s the correspondin g se t o f values :
f(X) = {f(x):xeX}.
Whether o r no t f~
l
exist s a s a function o n points, fo r a set Y C B, yo u ca n
always tak e th e inverse image
r1Y = {x:f(x)eY}.
For example , i f f(x) = sinx, the n
r
1
{ 0 } - { 0 , ± 7 r , ± 2 7 r , ± 3 7 r , . . . } .
Exercises 1
1. I s th e statemen t
If x e Q , the n x
2
e N
true o r fals e fo r th e followin g value s o f xl Why ?
a. x = 1/2.
b. x = 2.
c. x = \/2.
d. x = /2.
2. Fo r wha t rea l value s o f x i s th e convers e o f th e statemen t o f Exercis e 1
true?
3. Whic h o f the followin g statement s ar e tru e o f th e rea l numbers .
a. Fo r al l x ther e exist s a y suc h tha t y x
2.
b. Ther e exist s a y suc h tha t fo r al l x , y x
2.
c. Ther e exist s a y suc h tha t fo r al l x , y x
2.
d. Va , b, c, 3x suc h tha t ax
2
+ bx + c = 0 .
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