given f(x)=(5x+1)/4x and g(x)=[(√(x+1))-2/x^2-4.
find h(x), if (f.h)(x)=x
Answers
Answer:
Alan P.
Sep 20, 2015
Answer:
f
(
g
(
x
)
)
=
2
x
−
1
2
x
+
1
with Domain of
R
−
{
−
1
2
}
and Range of
R
−
{
1
}
Explanation:
It might help if we replace the variable
x
in
f
(
x
)
with a different variable than the one used in
g
(
x
)
. (We can do this because
x
is just an arbitrary place holder). Suppose for example we write:
XXX
f
(
w
)
=
1
w
+
1
Then it might be easier to see how we can replace
XXX
w
XXX
with
XXX
g
(
x
)
XXX
f
(
g
(
x
)
)
=
1
g
(
x
)
+
1
XXXXXXX
=
1
2
2
x
−
1
+
1
XXXXXXX
=
1
2
+
2
x
−
1
2
x
−
1
XXXXXXX
=
2
x
−
1
2
x
+
1
The
f
(
g
(
x
)
)
is defined for all Real values of
x
for which
XXX
2
x
+
1
≠
0
XXX
x
≠
−
1
2
That is, the Domain of
f
(
g
(
x
)
)
is
R
−
{
−
1
2
}
XXX
(or, if you prefer)
(
−
∞
,
−
1
2
)
∪
(
−
1
2
,
+
∞
)
One way to determine the range is to ask: "Is there any value,
c
for which
2
x
−
1
2
x
+
1
=
c
is impossible?"
XXX
2
x
−
1
=
c
(
2
x
+
1
)
XXX
2
x
−
2
c
x
=
c
+
1
XXX
x
=
c
+
1
2
(
c
−
1
)
This equation is clearly undefined if
c
=
1
Therefore the Range of
f
(
g
(
x
)
)
is
XXX
R
−
{
1
}
XXX XX
(...or,
(
−
∞
,
1
)
∪
(
1
,
+
∞
)
)
This can also be seen from the graph of
2
x
+
1
2
x
−
1
graph{(2x-1)/(2x+1) [-5.546, 5.55, -2.773, 2.774]}
Not sure what is meant by (f.h) (X). Multiplication of functions?
And then why is g(X) provided. Dummy info??