If y=(x^2+14x+9)/(x^2+2x+3) then what is the range of y ?

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Answer 1 · 2018-07-27T05:44:17

The range is #y in [-5, 4]#

The function is

#y=(x^2+14x+9)/(x^2+2x+3)#

#=>#, #y(x^2+2x+3)=(x^2+14x+9)#

#=>#, #yx^2+2yx+3y=x^2+14x+9#

#=>#, #yx^2-x^2+2yx-14x+3y-9=0#

#=>#, #(y-1)x^2+(2y-14)x+(3y-9)=0#

This is a quadratic equation in #x# and in order to have solutions, the discriminant #>=0#

#Delta=b^2-4ac>=0#

#(2y-14)^2-4(y-1)(3y-9)>=0#

#4y^2-56y+196-12y^2+48y-36>=0#

#8y^2+8y-160<=0#

#y^2+y-20<=0#

#(y+5)(y-4)<=0#

Solving this inequality with a sign chart or graphically yields

#y in [-5, 4]#

The range is #y in [-5, 4]#

graph{(x^2+14x+9)/(x^2+2x+3) [-10, 10, -5, 5]}