Problem

For \(f(x)~=~\sqrt{x~-~2}\) and \(g(x)~=~\ln(x~-~2)\), state \(\text{both}~\text{domains}~\text{and}~\text{ranges}\), then compare each pair using equality, proper-subset, or neither-containment relationships.

Square-root and logarithm graphs beginning near x equals 2 on shared axes.

Big Picture

What this problem is really about

Build each domain and range from its own defining restriction before comparing the sets. The square root permits its boundary input and produces only nonnegative outputs, while the logarithm requires a strictly positive argument but can produce any real output. Then use those boundary and output differences to determine the inclusion relationships.

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Preview only

Domain f=(−∞,∞); range f=[2,∞). Domain g=(−∞,∞); range g=(2,∞). Thus the domains are equal, while range g is a proper subset of range f.

Domain f=[2,∞); range f=[0,∞). Domain g=(2,∞); range g=(−∞,∞). Thus domain g is a proper subset of domain f, while range f is a proper subset of range g.

Domain f=[2,∞); range f=[0,∞). Domain g=[2,∞); range g=(−∞,∞). Thus the domains are equal, while range f is a proper subset of range g.

Domain f=(2,∞); range f=(−∞,∞). Domain g=[2,∞); range g=[0,∞). Thus domain f is a proper subset of domain g, while range g is a proper subset of range f.

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For \(f(x)~=~\sqrt{x~-~2}\) and \(g(x)~=~\ln(x~-~2)\), state \(\text{both}~\text{domains}~\text{and}~\text{ranges}\), then compare each pair using equality, proper-subset, or neither-containment relationships.

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