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At my school we are going over rational functions and discontinuities, and we often are asked to find the domain/range of a function. Sometimes the range feels quite lengthy, however, which is why I'm asking this question.

For example, look at the function

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

My teacher makes us write the domain of $f\left(x\right)$ as

$$\left(-\infty,-3\right)\cup\left(-3,-1\right)\cup\left(-1,2\right)\cup\left(2,\infty\right)$$

I would like to know if this is equivalent

$$\{x\in \mathbb{R},x\ne[-3,-1,2]\}$$

Even if this is not the same, is there a better way to write the domain?

Krypto14
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1 Answers1

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$\lbrace x\in \mathbb{R}, x\neq [-3,-1,2]\rbrace$ doesn't really make sense because $x$ is a real number, so it can't be equal to $[-3, -1, 2]$ regardless of what that notation means. If you want to use set notation like that, an alternative would be $$\lbrace x\in \mathbb{R} \mid x\not\in \lbrace -3, -1, 2 \rbrace\rbrace,$$ which is exactly what you were trying to express, but with proper set notation. Another way you could write it is $$x\in \mathbb{R}\setminus \lbrace -3, -1, 2\rbrace,$$

which means all of $\mathbb{R}$ without the set $\lbrace -3, -1, 2\rbrace$.

DMcMor
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