Problem

A control setting allows all real inputs, has neutral output \(0\), and changes smoothly from negative to positive output. State the function family, domain, neutral graph feature, and trend that fit this story.

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Family: reciprocal. Domain: excludes an asymptote input. Neutral feature: no defined origin crossing. Trend: separate branches approach horizontal and vertical asymptotes.

Family: saturation. Domain: typically one-sided. Neutral feature: no required sign-changing origin. Trend: increases toward a maximum-capacity asymptote.

Family: cube root. Domain: all real inputs. Neutral feature: x-intercept at the neutral setting. Trend: increases smoothly from negative to positive outputs.

Family: square root. Domain: starts at a delayed endpoint. Neutral feature: no required origin crossing. Trend: increases concave down on a one-sided domain.

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A control setting allows all real inputs, has neutral output \(0\), and changes smoothly from negative to positive output. State the function family, domain, neutral graph feature, and trend that fit this story.

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