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Byju's Answer
Standard XII
Mathematics
Properties of Iota
If f and ...
Question
If
f
and
g
are two functions such that
(
f
g
)
(
x
)
=
(
g
f
)
(
x
)
for all
x
. Then
f
and
g
may be defined as
A
f
(
x
)
=
√
x
,
g
(
x
)
=
cos
x
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B
f
(
x
)
=
x
3
,
g
(
x
)
=
x
+
1
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C
f
(
x
)
=
x
−
1
,
g
(
x
)
=
x
2
+
1
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D
f
(
x
)
=
x
m
,
g
(
x
)
=
x
n
where
m
,
n
are unequal integers
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Solution
The correct option is
D
f
(
x
)
=
x
m
,
g
(
x
)
=
x
n
where
m
,
n
are unequal integers
A
)
f
(
x
)
=
√
x
,
g
(
x
)
=
c
o
s
x
t
h
e
n
(
f
g
)
(
x
)
=
√
cos
x
a
n
d
(
g
f
)
(
x
)
=
cos
√
x
.They are not equal.
B)
f
(
x
)
=
x
3
,
g
(
x
)
=
x
+
1
t
h
e
n
(
f
g
)
(
x
)
=
(
x
+
1
)
3
a
n
d
(
g
f
)
(
x
)
=
x
3
+
1
.They are not equal.
C)
f
(
x
)
=
x
1
,
g
(
x
)
=
x
2
+
1
t
h
e
n
(
f
g
)
(
x
)
=
x
2
a
n
d
(
g
f
)
(
x
)
=
x
2
−
2
x
. They are not equal.
D)
f
(
x
)
=
x
m
,
g
(
x
)
=
x
n
t
h
e
n
(
f
g
)
(
x
)
=
x
n
+
m
a
n
d
(
g
f
)
(
x
)
=
x
n
+
m
.Hence
(
f
g
)
(
x
)
=
(
g
f
)
(
x
)
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Similar questions
Q.
If
g
(
f
(
x
)
)
=
∣
∣
sin
x
∣
∣
and
f
(
g
(
x
)
)
=
(
sin
√
x
)
2
, then