range and domain of root 3+x^2
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Step-by-step explanation:
Given:
y
=
√
x
−
3
x
2
The argument of the square root must be greater than or equal to 0:
x
−
3
x
2
≥
0
Becuase this quadratic is the equation of a parabola that opens downward we know that, if we find the roots of the quadratic, the quadratic will be greater than 0 between the roots.
x
−
3
x
2
=
0
x
(
1
−
3
x
)
=
0
x
=
0
and
x
=
1
3
This gives us the domain
0
≤
x
≤
1
3
We know that the minimum for the range occurs at either root:
0
≤
y
We know that the maximum for the range will occur halfway between the roots,
x
=
1
6
y
=
√
1
6
−
3
(
1
6
)
2
y
=
√
6
36
−
3
36
y
=
√
3
6
This makes the range become
0
≤
y
≤
√
3
6
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