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

1 Answer
Jul 20, 2017

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

Explanation:

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]}