A function f is defined by f: x→ x + 1. Another function g is defined such that gf : x → x² + 2x + 5. Express gf(x) in the form a(x + 1)² + b and hence write down the expression for g(x).
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A function f is defined by f: x→ x + 1. Another function g is defined such that gf : x → x² + 2x + 5. Express gf(x) in the form a(x + 1)² + b and hence write down the expression for g(x).
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Answer:
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Step-by-step explanation:
f
is given by
f
(
x
)
=
2
x
−
5
Then, we have
(i)
f
(
0
)
=
2
(
0
)
−
5
=
−
5
(ii)
f
(
7
)
=
2
(
7
)
−
5
=
9
(iii)
f
(
−
3
)
=
2
(
−
3
)
−
5
=
−
11
Answer:
Step-by-step explanation: Let's find
�
g by examining the composition
�
�
(
�
)
gf(x) and expressing it in the given form
�
(
�
+
1
)
2
+
�
a(x+1)
2
+b.
The function
�
f is defined as
�
:
�
↦
�
+
1
f:x↦x+1, and
�
g is defined as
�
�
:
�
↦
�
2
+
2
�
+
5
gf:x↦x
2
+2x+5.
Now, let's express
�
�
(
�
)
gf(x) in the form
�
(
�
+
1
)
2
+
�
a(x+1)
2
+b:
�
�
(
�
)
=
�
2
+
2
�
+
5
gf(x)=x
2
+2x+5
=
�
2
+
2
�
+
1
+
4
=x
2
+2x+1+4
=
(
�
+
1
)
2
+
4
=(x+1)
2
+4
Now, we can see that
�
�
(
�
)
gf(x) is in the form
�
(
�
+
1
)
2
+
�
a(x+1)
2
+b where
�
=
1
a=1 and
�
=
4
b=4.
Therefore,
�
�
(
�
)
=
(
�
+
1
)
2
+
4
gf(x)=(x+1)
2
+4.
Now, we know
�
�
(
�
)
gf(x), and we can express
�
(
�
)
g(x) by finding
�
g such that
�
:
�
↦
(
�
+
1
)
2
+
4
g:x↦(x+1)
2
+4.
So,
�
(
�
)
=
(
�
+
1
)
2
+
4
g(x)=(x+1)
2
+4.