How do you find the domain and range of #f(x)=(x^2+5x-6)/(x^2-5x+6)#?

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Answer 1 · 2017-07-20T17:14:02

The domain is #RR-{2,3}#
The range is #(-oo,-49]uu[-1,+oo)#

For the domain, the denominator must be #!=0#

Therefore,

The domain is #RR-{2,3}#

For the range, let rewrite the function

#y=(x^2+5x-6)/(x^2-5x+6)#

#y(x^2-5x+6)=x^2+5x-6#

#yx^2-x^2-5yx-5x+6y+6=0#

#(y-1)x^2-5(y+1)x+6(y+1)=0#....................#(1)#

Let 's calculate the discriminant of equation #(1)#

#Delta>=0#

#25(y+1)^2-24(y-1)(y+1)>=0#

#25(y^2+2y+1)-24(y^2-1)>=0#

#y^2+50y+49>=0#

#(y+49)(y+1)>=0#

The range is #(-oo,-49]uu[-1,+oo)#

graph{(x^2+5x-6)/(x^2-5x+6) [-132.6, 134.3, -94.2, 39.3]}