What is the range of the function f(x) = 1 / (x^2 - 2x + 3), where x is a real number?
Show the answer and explanation
Correct answer
A. (0, 1/2]
Principle or equation
The range of a rational function with a quadratic denominator can be found by analyzing the minimum of the denominator.
Why this answer is correct
x^2 - 2x + 3 = (x - 1)^2 + 2, which has minimum value 2. Thus f(x) = 1 / ((x-1)^2 + 2) has maximum 1/2 and approaches 0 as x → ±∞. Since the denominator is always positive, f(x) > 0. So range is (0, 1/2].
Example
For f(x) = 1/(x^2 + 1), 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.