Question from http://www.sgforums.com/forums/2297/topics/401082
Given that y = (x2 + 2x - c) / (x - 1) and x is real, find the range of values of c for which y can take all real values.
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Answer:
y = (x2 + 2x - c) / (x - 1)
y (x - 1 ) = x2 + 2x - c
yx - y = x2 + 2x - c
x2 + 2x - yx - c + y = 0
x2 + (2 - y) x + (y - c) = 0
Since x is real, discriminant ≥ 0
(2 - y)2 - 4 (1) (y - c) ≥ 0
4 - 4y + y2 - 4y + 4c ≥ 0
y2 - 8y + (4c + 4) ≥ 0
Since the equation needs to be ≥ 0, it means it has either equal roots or no real roots
so (-8)2 - 4 (1) (4c + 4) ≤ 0
64 - 16c - 16 ≤ 0
48 - 16c ≤ 0
48 ≤ 16c
c ≥ 3 (ans)
Graph example for clarity:
Plotted with
(i) c = 2
(ii) c = 4
(iii) c = 3
