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How can I find the domain of this function?

$$f(x)=\frac{x\sin(x)+\cos(x)}{1-\cos(x)} + \frac{|x|-2}{x^2-4}$$

I assume we don't want the denominator to be zero, but do we have to combine the denominators first?

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2 Answers 2

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Since combining the denominators is just the product of both denominators, we know that if either one is equal to zero, the function will be undefined there:

The function is undefined when:

  • $1-\cos x = 0 \implies \cos x = 1 \implies x = 0 \pm 2\pi k, \;\;k\in \mathbb Z$;
  • $x^2 - 4= 0 \implies x=2 \text{ or } x=-2$.
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Hint:

No, just find the values for which $1-\cos(x)=0$ and $x^2-4=0$. Those values of $x$ restrict the domain.

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