What is the range of the function f(x)=(3x^2+3x-6)/(x^2-x-12)f(x)=3x2+3x6x2x12?

2 Answers
Dec 30, 2017

The range is yin (-oo,0.614]uu[2.692,+oo)y(,0.614][2.692,+)

Explanation:

Let y=(3x^2+3x-6)/(x^2-x-12)y=3x2+3x6x2x12

To find the range, proceed as follows

y(x^2-x-12)=3x^2+3x-6y(x2x12)=3x2+3x6

yx^2-3x^2-yx-3x-12y+6=0yx23x2yx3x12y+6=0

x^2(y-3)-x(y+3)-(12y-6)=0x2(y3)x(y+3)(12y6)=0

This is a quadratic equation in xx and in order for this equation to have solutions, the discriminant Delta>=0

Delta=b^2-4ac=(-(y+3))^2-4(y-3)(-(12y-6))>=0

y^2+6y+9+4(y-3)(12y-6)>=0

y^2+6y+9+4(12y^2-42y+18)>=0

y^2+6y+9+48y^2-168y+72>=0

49y^2-162y+81>=0

y=(162+-sqrt(162^2-4*49*81))/(2*49)

=(162+-101.8)/(98)

Therefore,

The range is yin (-oo,0.614]uu[2.692,+oo)

graph{(3x^2+3x-6)/(x^2-x-12) [-14.24, 14.23, -7.12, 7.12]}

Dec 30, 2017

Range: f(x)in RR or (-oo,oo)

Explanation:

f(x) =(3x^2+3x-6)/(x^2-x-12) or

f(x) =(3(x+2)(x-1))/((x-4)(x+3))

f(x)=0 for (x=1, x=-2)

f(x) is undefined for (x=-3, x=4)

f(x)= oo and f(x) = -oo when x approaches -3 and 4

Therefore range is any real value ,i.e f(x)in RR or (-oo,oo)

Range: f(x)in RR or (-oo,oo)

graph{(3x^2+3x-6)/(x^2-x-12) [-40, 40, -20, 20]} [Ans]