f(x)= 2x+1 and g(f(x))=4x^2 +4x+4 find g(x) given that g(x)=ax^2+bx
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0
Answer:
g
(
x
)
=
x
2
+
2
Explanation:
We know that
f
(
x
)
=
2
x
+
1
and
g
(
x
)
=
a
x
2
+
b
x
+
c
.
⇒
g
(
f
(
x
)
)
=
a
(
2
x
+
1
)
2
+
b
(
2
x
+
1
)
+
c
Expanding the parentheses:
⇒
g
(
f
(
x
)
)
=
a
(
4
x
2
+
4
x
+
1
)
+
2
b
x
+
b
+
c
⇒
g
(
f
(
x
)
)
=
4
a
x
2
+
4
a
x
+
a
+
2
b
x
+
b
+
c
Rearranging the variables:
⇒
g
(
f
(
x
)
)
=
4
a
x
2
+
(
4
a
+
2
b
)
x
+
a
+
b
+
c
Now, we are also given the fact that
g
(
f
(
x
)
)
=
4
x
2
+
4
x
+
3
.
Comparing coefficients:
⇒
4
a
=
4
∴
a
=
1
and
⇒
4
a
+
2
b
=
4
⇒
4
×
(
1
)
+
2
b
=
4
⇒
4
+
2
b
=
4
⇒
2
b
=
0
∴
b
=
0
and
⇒
a
+
b
+
c
=
3
⇒
(
1
)
+
(
0
)
+
c
=
3
⇒
1
+
c
=
3
∴
c
=
2
Therefore,
g
(
x
)
=
(
1
)
x
2
+
(
0
)
x
+
(
2
)
=
x
2
+
2
.
Step-by-step explanation:
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