What is the domain for the function;

Question

BlackTom AI · Accepted Answer

To determine the domain of f(x) = 1/√(3 − x), we need the radicand to be positive (since square roots of negative numbers are not real) and we must avoid division by zero (the denominator cannot be zero). Thus, 3 − x > 0 implies x < 3, giving the domain as all real numbers less than 3. Option a: 'a. x ∈R \ x > 1' describes all real numbers except those greater than 1, i.e., x ≤ 1. This excludes values between negative infinity and 1, which is not the correct restriction for x < 3, and it includes many x ≥ 3 that would invalidate the radicand, so this option is incorrect. Option b: 'b. x ∈(1,[math: ∞] \infty )\{2}' is a garbled notation that attempts to express a subset of the real line starting after 1 and excluding 2. However, even if read as an interval (1, ∞) with 2 removed, it would include values greater than or equal to 3 and does not align with the needed x < 3 condition. Therefore this option does not correctly describe the domain. Option c: 'c. x ∈R \ 2' means all real numbers except x = 2. This would still include x ≥ 3, which are invalid due to the radicand (3 − x) becoming non-positive, so it is incorrect. Option d: 'd. x ∈R \ x > 2' is another ambiguous phrasing that seems to describe all real numbers except those where x > 2, effectively giving x ≤ 2. While this inadvertently excludes many x > 2, it still permits x values between 2 and 3 and above 3, depending on interpretation, and it does not properly enforce x < 3 as the only restriction. Hence this option is not correct. In summary, none of the presented options precisely state the correct domain, which should be x < 3 (i.e., (−∞, 3)). The key issue across all options is that they either mischaracterize the allowed x-values or fail to express the strict inequality required by the radicand being positive. The essential concept is that the square root requires a positive radicand and the result must avoid division by zero, leading to x < 3.

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