Question

Find the key number to the equation, the intervals into which the key numbers divide the number line, and the sign of y over each of these intervals. y=x+1/(x-1)^2

Answer 1

The function seems to be y = (x+1) / (x -1)^2

1) key number

The key number (also called critical values) is where the denominator equals zero.

(x -1)^2 = 0 => x = 1

2) intervals: (-∞,1) ,(1,∞)

3) Sign of the function

The sign of the denominator is always positive, then the sign of the function is the same sign of the numerator 

x +1 > 0 => x > - 1

x+ 1 < 0 => x < -1

Then in the interval (-∞,1) the function is negative for (-∞,-1) and positive for (-1,1).

The function is positive for all the values in the interval (1,∞)

Briefle, the functionis negative in (-∞,-1) and positive in (-1,∞).