Problem

For \(f(x)~=~x^2\) on all real \(x\), give fixed-input evidence and determine its one-to-one status, horizontal-line-test result, and whether its inverse relation is a function.

Coordinate graph of y equals x squared with a dashed horizontal line crossing the parabola at two marked points to support a horizontal-line-test check. — Check whether one horizontal line matches more than one point on the graph.

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Evidence: f(−1)=f(1)=1, so two inputs share an output. Conclusion: the function is not one-to-one, fails the horizontal-line test, and its inverse relation is not a function.

Evidence: every vertical line meets y=x² once. Conclusion: the function is one-to-one, passes the horizontal-line test, and its inverse relation is a function.

Evidence: the graph has a turning point at (0,0). Conclusion: the function is one-to-one on all real x, passes the horizontal-line test, and its inverse relation is a function.

Evidence: f(−1)=f(1)=1, so two inputs share an output. Conclusion: the function is one-to-one, passes the horizontal-line test, and its inverse relation is a function.

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For \(f(x)~=~x^2\) on all real \(x\), give fixed-input evidence and determine its one-to-one status, horizontal-line-test result, and whether its inverse relation is a function.

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