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

## ${f}_{x} = \left(\sin \frac{x}{{x}^{3} + x - 2}\right) - \left(\ln \frac{x + 1}{x}\right)$

Jan 25, 2018

${D}_{f} = \left(- 1 , 0\right) \cup \left(0 , 1\right) \cup \left(1 , + \infty\right)$

#### Explanation:

For $f$ to be defined in $\mathbb{R}$ you need

• ${x}^{3} + x - 2 \ne 0$

• $x + 1 > 0$

• $x \ne 0$

With Horner method to factor ${x}^{3} + x - 2$ you get

${x}^{3} + x - 2 = \left(x - 1\right) \left({x}^{2} + x + 2\right)$

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 $x - 1 \ne 0 \iff x \ne 1$

So you need

• $x \ne 1$
• $x >$$- 1$

• $x \ne 0$

Therefore the domain of $f$ will be

${D}_{f} = \left(- 1 , 0\right) \cup \left(0 , 1\right) \cup \left(1 , + \infty\right)$