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Graph

The graph of a function is the representation of the function using the Cartesian coordinate system. It is the collection of all points $$$\big(x,f(x)\big)$$$ where $$x$$ ranges through $$D_f$$.

 The graph of $$x\mapsto 1/(x-1)^2.$$
The graph of $$x\mapsto 1/(x-1)^2.$$

Because $$f(x)$$ takes only one value, each $$x$$-coordinate in the graph has only one corresponding $$y$$-coordinate. Any vertical line intersects the graph at most once.

The blue circle on the left is not the graph of a function since the red vertical line intersects it twice. The semi-circle on the right is the graph of a function.
The blue circle on the left is not the graph of a function since the red vertical line intersects it twice. The semi-circle on the right is the graph of a function.