Find range of f(x)=x^2/x^4+1
Answers
Answer:The domain is(−∞,∞) and the range [ 0 , 1 /2 ]
Given:
f
(
x
)
=
x
2
1
+
x
4
Note that for any real value of
x
, the denominator
1
+
x
4
is non-zero.
Hence
f
(
x
)
is well defined for any real value of
x
and its domain is
(
−
∞
,
∞
)
.
To determine the range, let:
y
=
f
(
x
)
=
x
2
1
+
x
4
Multiply both ends by
1
+
x
4
to get:
y
x
4
+
y
=
x
2
Subtracting
x
2
from both sides, we can rewrite this as:
y
(
x
2
)
2
−
(
x
2
)
+
y
=
0
This will only have real solutions if its discriminant is non-negative. Putting
a
=
y
,
b
=
−
1
and
c
=
y
, the discriminant
Δ
is given by:
Δ
=
b
2
−
4
a
c
=
(
−
1
)
2
−
4
(
y
)
(
y
)
=
1
−
4
y
2
So we require:
1
−
4
y
2
≥
0
Hence:
y
2
≤
1
4
So
−
1
2
≤
y
≤
1
2
In addition note that
f
(
x
)
≥
0
for all real values of
x
.
Hence
0
≤
y
≤
1
2
So the range of
f
(
x
)
is
[
0
,
1
2
]