What is the range of the function \(f(x)=\frac{1}{|x|+1}\)?

Correct answer: \((0,1]\)

For every real \(x\), \(|x|\ge 0\), so \(|x|+1\ge 1\), which gives \(0<f(x)\le 1\). At \(x=0\), the function equals 1. As \(|x|\) increases, the function approaches 0 but never becomes 0. Hence, the range is \((0,1]\). Option B is incorrect because 0 is not attained. Exam tip: For a reciprocal function, identify the minimum and possible limiting values of the denominator to determine the range.