How do we find the range of an algebraic rational function of the form a1x2+b1x+c1a2x2+b2x+c2for real values of x?

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dizzydevila6cm · Accepted Answer

Step 1Lety=16x2−12x+916x2+12x+9Since the denominator is non zero, we can multiply by the denominator and shift terms to getx2(16y−16)+x(12y+12)+(9y−9)=0This is a quadratic equation with real roots, which means that the discriminant is non-negative. Hence,0≤(12y+12)2−(16y−16)(9y−9)=−144(3−y)(1−3y),so the range is contained within [13, 3]Note that for this to truly be a quadratic, we require 16y−16≠0, or that y≠1. In this case, we&#039;re solving the linear equation 24x=0, which has a solution x=0, so y=1 is in the range. Henceforth, let&#039;s ignore y=1 as an edge case.Conversely, for any y∈[13, 3], we can solve the corresponding quadratic equation since the discriminant is non-negative, to obtain a real x. Hence, that y value is in the range.Hence, the range is exactly [13, 3]Step 2Note: This works for rational functions where the denominator is a quadratic with no real roots.It doesn&#039;t easily extend to higher degree polynomials.

How do we find the range of an

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