Question

How can i find the Domain of this f(x)?

`f_x=(sinx/(x^3+x-2))-(ln(x+1)/x)`

Answer 1

`D_f=(-1,0)uu(0,1)uu(1,+oo)`

Explanation:

For `f` to be defined in `RR` you need

• `x^3+x-2!=0`

• `x+1>0`

• `x!=0`

With Horner method to factor `x^3+x-2` you get

`x^3+x-2= (x-1)(x^2+x+2)`

But `x^2+x+2` is always `>0` because the discriminant `Δ<0`
`Δ=b^2-4ac=1-4*2*1=1-8=-7<0`

As a result you need `x-1!=0 <=> x!=1`

So you need

• `x!=1`

• `x>` `-1`

• `x!=0`

Therefore the domain of `f` will be

`D_f=(-1,0)uu(0,1)uu(1,+oo)`