How do you find the domain and range of f(x)= (2-x)/(x^2+7x+12)f(x)=2−xx2+7x+12?

1 Answer
Narad T.
Jan 28, 2017

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

Explanation:

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[

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