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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
1:DOM 1 xf 1:RAN 11 xff
iii. 0AND205 xx,xxf
205LET xy2
520
y
x
2
1
520
x
xf
0:DOM xf 20:RAN xff
20:DOM 1 xf 0:RAN 11 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