Problem

For \(f(x)~=~e^x\) and inverse \(\ln(x)\), state both functions' domains and ranges and explain how the sets exchange.

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Original eˣ: domain (−∞,∞), range (0,∞). Inverse ln(x): domain (0,∞), range (−∞,∞). This correctly exchanges the two sets.

Original eˣ: domain (−∞,∞), range (0,∞). Inverse ln(x): domain (−∞,∞), range (0,∞). This copies the sets instead of exchanging them.

Original eˣ: domain (−∞,∞), range [0,∞). Inverse ln(x): domain [0,∞), range (−∞,∞). This incorrectly includes zero in the exponential range and logarithm domain.

Original eˣ: domain (−∞,∞), range (0,∞). Inverse ln(x): domain (0,∞), range (0,∞). This exchanges the domain but incorrectly keeps the original range.

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For \(f(x)~=~e^x\) and inverse \(\ln(x)\), state both functions' domains and ranges and explain how the sets exchange.

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