Problem

Compare \(A(x)~=~8~+~500/x\) and \(B(x)~=~10~+~300/x\). State each domain, asymptotes, \(x\text{-}\text{intercept}\), and monotonic behavior on every branch.

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What this problem is really about

Build the same feature inventory for both functions before comparing them: denominator restrictions and vertical asymptotes, long-run levels, zeros, then behavior on each connected branch. Their shared reciprocal structure explains several similarities, while the different constants control the features that separate them. Keeping parallel rows prevents a value from one model from drifting into the other.

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Domain: both exclude x=0. Asymptotes: both have vertical x=0; A has horizontal y=8 and B has horizontal y=10. Intercepts: A has x=−62.5 and B has x=−30. Behavior: both decrease on each branch.

Domain: both include x=0. Asymptotes: both have horizontal x=0; A has vertical y=8 and B has vertical y=10. Intercepts: neither has an x-intercept. Behavior: both increase on each branch.

Domain: both are all real. Asymptotes: neither has a vertical asymptote; A has horizontal y=500 and B has horizontal y=300. Intercepts: both have x=0. Behavior: both increase.

Domain: both exclude x=0. Asymptotes: both have vertical x=0 and horizontal y=8. Intercepts: both have x=−62.5. Behavior: their formulas produce identical branches.

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Compare \(A(x)~=~8~+~500/x\) and \(B(x)~=~10~+~300/x\). State each domain, asymptotes, \(x\text{-}\text{intercept}\), and monotonic behavior on every branch.

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