SOLUTION: How do i find the range of f(x)={{{1/(x^2+3x+5)}}}

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Question 760261: How do i find the range of f(x)=1%2F%28x%5E2%2B3x%2B5%29
Answer by reviewermath(1029) About Me  (Show Source):
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Q:
How do i find the range of f(x)=1%2F%28x%5E2%2B3x%2B5%29
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A:
Let y = 1%2F%28x%5E2%2B3x%2B5%29
Solve for x in terms of y.
x%5E2+%2B+3x+%2B+5 = 1%2Fy
x%5E2+%2B+3x+%2B+%285+-+1%2Fy%29+=+0
Use the Quadratic Formula to solve for x.
x+=+%28-3+%2B-+sqrt%28+3%5E2-4%2A1%2A%285+-+1%2Fy%29+%29%29%2F%282%2A1%29+
x+=+%28-3+%2B-+sqrt%28+-11+%2B+4%2Fy+%29%29%2F2+
The expression inside the square root is -11+%2B+4%2Fy.
-11+%2B+4%2Fy ≥ 0
%284+-+11y%29%2Fy ≥ 0
Using Sign Test
0 < y ≤ 4%2F11
In interval notation, the range is (0, 4/11].