How do you find the domain and range of #f(x)= (2-x)/(x^2+7x+12)#?

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Answer 1 · 2017-01-28T14:55:05

The domain is #D_f(x)=RR-{-4,-3}#
The range is # ]-oo, -21.95]uu[-0.455,+oo[#

Let's factorise the denominator

#x^2+7x+12=(x+4)(x+3)#

As you cannot divide by #O#, #x!=-4# and #x!=-3#

The domain of #f(x)# is #D_f(x)=RR-{-4,-3}#

Let #y=(2-x)/(x^2+7x+12)#

Then,

#yx^2+7yx+12y=2-x#

#yx^2+7yx+x+12y-2=0#

#yx^2+(7y+1)x+12y-2#

Solving for #x#

The discriminant is

#Delta=b^2-4ac#

#=(7y+1)^2-4*y*(12y-2)#

#=49y^2+14y+1-48y^2+8y#

#=y^2+22y+1#

This has to be #>=0#

Therefore,

#22^2-4*1=484-4=480#

So,

#y=(-22+-sqrt480)/2#

#=-11+-2sqrt30#

Therefore,

#y in ]-oo, -21.95]uu[-0.455,+oo[#

The range is #f(x) in ]-oo, -21.95]uu[-0.455,+oo[#