What is the range of the function f(x) = 1 / (x^2 + 2)?
Show the answer and explanation
Correct answer
A. (0, 1/2]
Principle or equation
For a rational function with denominator always positive, the range is determined by the minimum and maximum of the denominator. Since x^2 ≥ 0, x^2 + 2 ≥ 2, so 1/(x^2+2) ≤ 1/2 and > 0.
Why this answer is correct
The denominator x^2 + 2 is always positive and has a minimum value of 2 at x = 0. Therefore, the function attains its maximum value 1/2 at x = 0. As |x| increases, the denominator grows without bound, so the function approaches 0 but never reaches it. Thus the range is (0, 1/2].
Example
For f(x) = 1/(x^2+1), denominator ≥1, so range is (0,1].
This published item includes a stored explanation and passed the platform’s publication workflow. It is independent preparation material, not a claim of an official or recalled examination question.