find the range of y= x^2-2x+9÷x^2+2x+9
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Explanation:
The function is
y
=
x
2
+
14
x
+
9
x
2
+
2
x
+
3
⇒
,
y
(
x
2
+
2
x
+
3
)
=
(
x
2
+
14
x
+
9
)
⇒
,
y
x
2
+
2
y
x
+
3
y
=
x
2
+
14
x
+
9
⇒
,
y
x
2
−
x
2
+
2
y
x
−
14
x
+
3
y
−
9
=
0
⇒
,
(
y
−
1
)
x
2
+
(
2
y
−
14
)
x
+
(
3
y
−
9
)
=
0
This is a quadratic equation in
x
and in order to have solutions, the discriminant
≥
0
Δ
=
b
2
−
4
a
c
≥
0
(
2
y
−
14
)
2
−
4
(
y
−
1
)
(
3
y
−
9
)
≥
0
4
y
2
−
56
y
+
196
−
12
y
2
+
48
y
−
36
≥
0
8
y
2
+
8
y
−
160
≤
0
y
2
+
y
−
20
≤
0
(
y
+
5
)
(
y
−
4
)
≤
0
Solving this inequality with a sign chart or graphically yields
y
∈
[
−
5
,
4
]
The range is
y
∈
[
−
5
,
4
]
graph{(x^2+14x+9)/(x^2+2x+3) [-10, 10, -5, 5]}
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