Problem

For average cost, let \(f(x)~=~500/x\) be allocated fixed cost per item and \(g(x)~=~8\) be variable cost per item, with positive whole-number \(x\). Form \((f~+~g)(x)\), state the contextual domain, and identify and justify the long-run graph feature.

Neutral average-cost information card showing f of x equals 500 over x and g of x equals 8, without a graph, domain conclusion, or feature.

Big Picture

What this problem is really about

Build the combined model only after checking that the two components have compatible output units. Then keep the contextual domain beside the algebra: an item count is discrete and positive, not just any nonzero real. For the long-run feature, separate the shrinking reciprocal term from the constant term; their roles reveal the horizontal behavior without relying on a sketch.

Answer choices

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Preview only

(f + g)(x) = 500/(x + 8), so the average cost approaches 0 as x grows.

(f + g)(x) = 500/x + 8. For this context, x is a positive whole number, and the horizontal asymptote is x = 8.

(f + g)(x) = 500/x + 8. The graph has horizontal asymptote y = 8, but the contextual domain includes negative and fractional item counts.

(f + g)(x) = 500/x + 8. For this context, x is a positive whole number, and the horizontal asymptote is y = 8 because 500/x approaches 0 as x grows.

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For average cost, let \(f(x)~=~500/x\) be allocated fixed cost per item and \(g(x)~=~8\) be variable cost per item, with positive whole-number \(x\). Form \((f~+~g)(x)\), state the contextual domain, and identify and justify the long-run graph feature.

What this problem is really about

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