How do I find the range of f(x) = (x^2 + x + 1)/(x^2 - x + 1)?
I need to find the range of this rational function for JEE. I tried substituting some values but that does not give the complete picture. Is there a standard method for this type?
1 Answer
VNVikram Nair
✓ Accepted
· 15d ago
▲ 6
Let y = (x^2+x+1)/(x^2-x+1). Cross multiply: y(x^2-x+1) = x^2+x+1. Rearrange: (y-1)x^2 + (-y-1)x + (y-1) = 0. For real x, discriminant >= 0: (-y-1)^2 - 4(y-1)^2 >= 0. Expand: (y+1)^2 - 4(y-1)^2 >= 0. Simplify: -3y^2 + 10y - 3 >= 0, so 3y^2 - 10y + 3 <= 0, giving (3y-1)(y-3) <= 0. Therefore 1/3 <= y <= 3. Range is [1/3, 3].
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