SOLUTION: s(x) = 2x − 4 over x^2 + x − 2 State the domain and range. (Enter your answers using interval notation.)

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Question 1023119: s(x) = 2x − 4 over x^2 + x − 2
State the domain and range. (Enter your answers using interval notation.)
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
%282x-4%29%2F%28x%5E2%2Bx-2%29+=+%282x-4%29%2F%28%28x-1%29%28x%2B2%29%29
==> Domain of s(x) is (-infinity, -2)u(-2,1)u(1,infinity).
To find the range of s(x), let y+=+%282x-4%29%2F%28x%5E2%2Bx-2%29
==> yx%5E2%2Byx-2y+=+2x-4
==> yx%5E2%2B%28y-2%29x%2B%284-2y%29+=+0, a quadratic equation in x with coefficients in terms of y. For such expression to have real roots, the discriminant
b%5E2+-+4ac+=+%28y-2%29%5E2+-+4y%284-2y%29+%3E=0
<==> %28y-2%29%5E2+%2B8y%28y-2%29+%3E=0
<==> %28y-2%29%289y-2%29+%3E=0.
The solution to this inequality is the union (-infinity, 2/9]u[2, infinity) .
Incidentally this is also the range of s(x).