Question

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

Answer 1

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[`