What is the range of the function f(x) = 1 / (x^2 + 1)?
Show the answer and explanation
Correct answer
A. (0, 1]
Principle or equation
The range of a function is the set of all possible output values. For f(x) = 1/(x^2 + 1), since x^2 ≥ 0, the denominator is at least 1, so the fraction is at most 1. As x^2 grows, the fraction approaches 0 but never reaches 0. Thus the range is all positive numbers up to and including 1.
Why this answer is correct
x^2 + 1 ≥ 1, so 1/(x^2 + 1) ≤ 1. Also, since x^2 + 1 > 0, the fraction is always positive. When x = 0, f(0) = 1. As |x| increases, the value approaches 0 but never equals 0. Therefore the range is (0, 1].
Example
For g(x) = 1/(x^2 + 4), the range is (0, 1/4] because the denominator is at least 4.
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