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

Question

$f_{x} = (\frac{sin x}{x^{3} + x - 2}) - (\frac{ln (x + 1)}{x})$ $f_x=(sinx/(x^3+x-2))-(ln(x+1)/x)$

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Answer 1 · 2018-01-25T19:29:43

$D_{f} = (- 1, 0) \cup (0, 1) \cup (1, + \infty)$ $D_f=(-1,0)uu(0,1)uu(1,+oo)$

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

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

Therefore the domain of $f$ will be

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