function

IINNVVEERRSSEEFUNCTIONSFUNCTIONS

DOMAIN OF f

RA

NG

E O

F f

RANGE OF f -1

DO

MA

IN O

F f- -1

xx

y y

xx

yyf

1f

1 TO 1 FUNCTION

f(x) = y

f -1(y) = x

Dom(f)=Ran(f -1)

Ran(f)=Dom(f -1)

INVERSE FUNCTION

EXAMPLE 1GIVEN THAT THE FOLLOWING

FUNCTIONS HAS DOMAIN R. DETERMINE

WHETHER INVERSE FUNCTION EXIST

OR NOT.

i. ii.

iii. iv.

v. vi.

14 xx:f 13 2 xx:f

xx:f sin 13 xx:f2xx:f

7xx:f

i. YES (1 TO 1 FUNCTION)

ii. NO (MANY TO 1 RELATION)

iii. NO (MANY TO 1 RELATION)

iv. YES (1 TO 1 FUNCTION)

v. NO (MANY TO 1 RELATION)

vi. YES (1 TO 1 FUNCTION)

EXAMPLE 2DETERMINE WHICH OF THE FOLLOWINGFUNCTION HAS INVERSE FUNCTIONS IN THE SPECIFIED DOMAINS.i.ii.iii.iv.v.

0542 x,xxx:f

012 2 x,xx:f

0 x,xx:f

oo 0180cos x,xx:f

0542 x,xxx:f

i. YES (1 TO 1 FUNCTION)

ii. YES (1 TO 1 FUNCTION)

iii. YES (1 TO 1 FUNCTION)

iv. YES (1 TO 1 FUNCTION)

v. NO (MANY TO 1 FUNCTION)

EXAMPLE 3DETERMINE THE DOMAIN OF THE FUNCTION SO THAT AN INVERSE FUNCTION EXISTS.

xxx:f 52

xxxf 52 42525 2 xxf

25x 25xOR

EXAMPLE 4FIND THE INVERSE FUNCTION FOR

THEFOLLOWING FUNCTIONS.i.ii.

iii.

186 xx:f

2AND142 xx,xxx:f

34

4312

x,

xx

x:f

i.

ii.

186 xxf18-6LET xy

618y

x

6181 x

xf

34

4312

x,

xx

xf

4312

LET

xx

y

1243 xxy

1243 xyxy

1423 yxxy

1423 yyx

2314

yy

x

32

23141

x,

xx

xf

iii. 2AND142 xx,xxxf

3AND321 xx,xxf

2SINCE x

14LET 2 xxy

yx 32

yx 32

124

24

422

2

xxy

32 2 xy

32 yx

32 2 yx

EXAMPLE 5DETERMINE THE RANGE OF FIND THE INVERSE FUNCTION FOR THEFOLLOWING FUNCTIONS AND STATE ITS DOMAIN AND RANGE.

i.ii.iii.

0AND155 xx,xx:f

1AND22 xx,xxx:f

0AND205 xx,xx:f

.f

i. 0AND155 xx,xxf

155LET xy

515 y

x

5

151 xxf

0:DOM xf 15:RAN xff

15:DOM 1 xf 0:RAN 11 xff

Dom(f)=Ran(f -1)

Ran(f)=Dom(f -1)

ii.

222

22

22

2

xxy

1AND22 xx,xxxf

1:DOM xf 1:RAN xff

xxy 2LET 2

11 2 xy

11 yx1SINCE x

111 xxf


iii. 0AND205 xx,xxf

205LET xy2

520

y

x

2

1

520

x

xf

0:DOM xf 20:RAN xff


IF IS 1-1 FUNCTION, THE GRAPHS

AND ARE REFLECTIONS OF EACH OTHER IN THE LINE

.xy xfy xfy 1

f

x

yxy

xfy

xfy 1

(m,n)

(n,m)

EXAMPLE 6GIVEN THAT FIND ITS INVERSE AND SKETCH BOTH GRAPHS IN THE SAME DIAGRAM.

4AND43 xx,xxf

4AND43 xx,xxf 43 21 xxf

4:DOM xf 3:RAN xff 3:DOM 1 xf 4:RAN 11 xff

x

yf

1f

(3,-4)

(-4,3)

xy

INVERSE FUNCTION

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