Mathematics

Mathematics

Problem

For the function f(x)=x+42−xf(x)=\sqrt{\dfrac{x+4}{2-x}}, which of the following sets represents its domain? a) [−4,∞)[-4,\infty) b) [−4,2)[-4,2) c) [−4,2][-4,2] d) (−∞,2)(-\infty,2)
Intuitive explanation

The domain of a square root function requires the expression inside to be non-negative, but here you have a fraction under the radical, so you need both the numerator and denominator to satisfy specific conditions. The key insight is to set up a rational inequality: the fraction x+42−x\frac{x+4}{2-x} must be ≥0\geq 0. Use a sign chart to find where the numerator and denominator have compatible signs, remembering that the denominator can never equal zero.

Intuitive explanation

The domain of a square root function requires the expression inside to be non-negative, but here you have a fraction under the radical, so you need both the numerator and denominator to satisfy specific conditions. The key insight is to set up a rational inequality: the fraction x+42−x\frac{x+4}{2-x} must be ≥0\geq 0. Use a sign chart to find where the numerator and denominator have compatible signs, remembering that the denominator can never equal zero.

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