1:y =(x+ 1)/(x+2)=[(x+2)- 1]/(x+2)= 1- 1/(x+2)
If x is an arbitrary real number, 1/(x+2) is an arbitrary real number, and the range of values is (negative infinity, 1) and (1, positive infinity).
2:y=(3x- 1)/(2x+ 1) is also modified as above.
y =(3x- 1)/(2x+ 1)= 3/2 *(x- 1/3)/(x+ 1/2)= 3/2 *[(x+ 1/2)-5/6]/(x+ 1/2)
= 3/2 *[ 1-5/6(x+ 1/2)]
The range is any real number that is not equal to 3/2 {or (negative infinity, 3/2) and (3/2, positive infinity)}.
3:y=( 1-x^2)/( 1+x^2)=[( 1+x^2)-2x^2]/( 1+x^2)= 1-2x^2/( 1+x^2)
The numerator of 2x 2/( 1+x 2) fraction, and the denominator is divided by x 2 at the same time.
y= 1-2x^2/( 1+x^2)= 1-2/( 1+ 1/x^2)
x^2>； 0
1/x^2>； 0
1+ 1/x^2>； 1
0 & lt2/( 1+ 1/x^2)<； 2
-2 & lt； -2/( 1+ 1/x^2)<； 0
- 1 & lt； 1-2/( 1+ 1/x^2)<； 1
The range is (-1, 1).