The Graph of a Function Defined Rigorously
This animation builds the formal definition of a function's graph as the set of all points (x, f(x)). It demonstrates that a point M(x, y) lies on the graph exactly when y equals f(x), using the definition as a biconditional. A contrasting point N(x, y) with y different from f(x) is shown to lie off the curve, reinforcing why the graph is a precise geometric encoding of the function rule. Useful for students transitioning from informal curve-sketching to rigorous definitions.
16:9 · every frame verified for overlaps, spacing and edges before rendering
The prompt that made it
Explain rigorously the concept of the graph of a function f. Show that a point M(x,y) belongs to the graph if and only if y=f(x). Then show a point N(x,y) with y different from f(x) does not belong to the graph.
Make the next one in this series
- Show why the vertical line test follows directly from the definition of a function's graph
- Illustrate the graph of a piecewise function using the same point-membership argument
- Extend this reasoning to define and visualize the graph of a relation that is not a function
