How do you show that a function is negative?f(x)=2x/(x^2-1)
Given `f(x)=(2x)/(x^2-1)` , show where the function is negative.
(1) Like fractions, rational functions are negative if the numerator and denominator have opposite signs.
(2) The numerator: 2x<0 if x<0; 2x>0 if x>0
(3) The denominator: `x^2-1<0 => x^2<1 => -1<x<1` and `x^2-1>0 => x^2>1 => x<-1` or `x>1` .
(4) On the interval x<-1, the numerator is negative and the denominator is positive so f(x)<0
On the interval `-1<x<0` the numerator is negative and the denominator is positive so f(x)>0
On the interval 0<x<1 the numerator is positive and the denominator is negative so f(x)<0
On the interval x>1 the numerator and denominator are positive so f(x)>0
Thus f(x)<0 when x<-1 and 0<x<1.