Why is that the graph of f(x) = x is the straight line that is the bisector of the first quadrant? (or, amounting to the same thing, the bisector of the third quadrant)
By calculating the outputs for a few inputs, one can only plot a finite number of points, so it's not possible to prove it this way.
I thought of thinking of a point that satisfies y = x as the third vertice of a right triangle with congruent legs, and therefore by trigonometry the tangent is 1, so the other angles of this moving right triangle are always 45º, that is half the angle formed by the x and y axis, and therefore it's the angle bisector... But I'm confused about this.
I also thought that whatever the graph is, it must be the same if I reflect it through the bisector of the first quadrant (one can interchange x and y and the formula is the same), and that this is satisfied by the angle bisector itself, but also by others curves, so it doesn't prove.
I'm extremely stupid. Please don't make fun of me. I'm trying hard to understand this, so much that it has caused me a headache...