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Identify the graph of a rational function that is decreasing on the interval (–5, 5).
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GPT-5 mini
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One simple choice is
f(x) = 1/(x + 6).
Compute f'(x) = −1/(x + 6)^2, which is strictly negative for every x ≠ −6. Since the interval (−5, 5) does not contain −6, f is defined on (−5,5) and f'(x) < 0 there, so f is decreasing on (−5,5).
(Other examples also work, e.g. f(x) = −x is a polynomial rational function decreasing on the whole real line.)
f(x) = 1/(x + 6).
Compute f'(x) = −1/(x + 6)^2, which is strictly negative for every x ≠ −6. Since the interval (−5, 5) does not contain −6, f is defined on (−5,5) and f'(x) < 0 there, so f is decreasing on (−5,5).
(Other examples also work, e.g. f(x) = −x is a polynomial rational function decreasing on the whole real line.)
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