Problem

For \(f(x)~=~1/x\) and \(g(x)~=~x^2\), state \(\text{both}~\text{domains}~\text{and}~\text{ranges}\), then compare each pair using equality, proper-subset, or neither-containment relationships and give witnesses when needed.

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Domain f=ℝ{0; range f=ℝ{0. Domain g=ℝ; range g=[0,∞). Domain f is a proper subset of domain g, and range g is a proper subset of range f.

Domain f=ℝ; range f=ℝ. Domain g=ℝ{0; range g=(0,∞). Domain g is a proper subset of domain f, and range g is a proper subset of range f.

Domain f=ℝ{0; range f=ℝ{0. Domain g=ℝ; range g=[0,∞). Domain f is a proper subset of domain g, while neither range contains the other: −1 occurs only for f and 0 only for g.

Domain f=ℝ{0; range f=(0,∞). Domain g=ℝ; range g=[0,∞). Domain f is a proper subset of domain g, and range f is a proper subset of range g.

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For \(f(x)~=~1/x\) and \(g(x)~=~x^2\), state \(\text{both}~\text{domains}~\text{and}~\text{ranges}\), then compare each pair using equality, proper-subset, or neither-containment relationships and give witnesses when needed.

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