I’ve been staring at this problem for 10 minutes and still have no idea

9514 1404 393

  B.  (-∞, 1) ∪ (1, 2]

Step-by-step explanation:

When you are looking for answers to domain questions, you are looking for values of the variable(s) that make the function undefined. Here, there is a square root involved, and the rational function has a denominator that might be zero.

The function is only defined for square roots of non-negative numbers. That is ...

  2-x ≥ 0   ⇒   x ≤ 2

The rational function is only defined for non-zero denominators, so ...

  x-1 ≠ 0

  x ≠ 1

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So, you're looking for a domain description that includes all numbers less than or equal to 2, and excludes x=1. The attached number line shows a graph of this.

The domain divides into two parts: all numbers less than 1, and those numbers between 1 and 2 (including 2). This will be the union ...

  (-∞, 1) ∪ (1, 2]