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Good explanation. Another way to think about it that helps me (though if it doesn't help you, probably best just to forget about it) is to consider the result if I graphed the function. So if the x axis is x, and the y axis is f(x), then for a valid function, there should be no place along the x axis where I could draw a vertical line directly down or up and have that line intersect the function's curve/line in more than one place.
Using the examples given above, if you graphed the valid function outputs, you would see that there is only one output for each x input (i.e. only one spot on the graph for each x), and thus I cannot draw a vertical line intercepting the function in more than one place. That is NOT the case however with the invalid input, as I could draw a line and intercept the function's output for 2 at two places.
I hope this helps to clarify things for someone :)