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在解数学题时,常常先构建一元二次方程,用判别式的性质讨论一元二次方程根的情况来解题的方法叫判别式法,它应用十分广泛,现举例说明.一、求分式函数的值域例1求函数y=(x2+1)/(x2-x+1)的值域.解∵x2-x+1=(x-1/2)2+3/4>0恒成立,∴x∈R,原函数变形为(y-1)x2-yx+(y-1)=0.当y≠1时,方程为x的一元二次方程,∵x∈R,∴Δ≥0,即Δ=y2-4(y-1)2≥0,解得2/3≤y≤2.注意到y=1∈[2/3,2],故函数的值域为[2/3,2].
In solving mathematical problems, often first construct a quadratic equation, the nature of the discriminant to discuss the quadratic equation root equation to solve the problem is called discriminant method, it is widely used, are an example. Example 1 Find the range of the function y = (x2 + 1) / (x2-x + 1). Solution ∵x2-x + 1 = (x-1/2) 2 + 3/4> 0成x∈R, the transformation of the original function is (y-1) x2-yx + (y-1) = 0. When y ≠ 1, the equation is a quadratic equation of x, ∵x∈R, ∴Δ ≥ 0, that is, Δ = y2-4 (y-1) 2≥0, and solve 2 / 3≤y≤2. Notice that y = 1∈ [2/3,2], so the range of the function is [2 / 3,2].