Problem

An invertible function \(f\) has domain \([2,~\infty)\) and range \([-1,~\infty)\). State the inverse domain and range and explain the exchange.

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What this problem is really about

An inverse reverses every input-output pair, so the original output set becomes the inverse input set and the original input set becomes the inverse output set. Write all four fields explicitly to avoid copying intervals into the same roles. Preserve endpoint inclusion during the exchange, since an attained boundary value remains part of the corresponding reflected set.

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Original: domain [2,∞), range [−1,∞). Inverse: domain [2,∞), range [−1,∞). This copies the sets instead of exchanging output and input.

Original: domain [2,∞), range [−1,∞). Inverse: domain [−1,∞), range [−1,∞). This exchanges the domain but incorrectly keeps the original range.

Original: domain (2,∞), range (−1,∞). Inverse: domain (−1,∞), range (2,∞). This exchanges the sets but incorrectly removes both endpoints.

Original: domain [2,∞), range [−1,∞). Inverse: domain [−1,∞), range [2,∞). This correctly exchanges the original output and input sets.

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An invertible function \(f\) has domain \([2,~\infty)\) and range \([-1,~\infty)\). State the inverse domain and range and explain the exchange.

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