Why can't this be the graph of any function?

Question

I have seen that this can be the graph of an equation-

But I learn't that this can't be the graph of any equation-

Why is this?Why does this graph not exist?Please explain this to me.

Thanks for any help!!

Answer 1

It is the graph of an equation, it just isn't the graph of a $\mathbf{function}$. Your top graph is roughly the graph of $f(x)=-\sin(x)$, while the bottom graph could roughly be described as the equation $x=\sin(y)$.

Equations can take any form, but functions are only functions when there is a single y-value associated with every x-value. This can be described as the "vertical line test." If a vertical line goes through the graph of a "function" more than once, the the "function" isn't actually a function.

Answer 2

The convention is that the horizontal axis represents the domain of the function, and going up vertically from any point in the domain, you will intersect the graph at a "height" representing the value of the function at that point in the domain. Now, a function by definition must have a SINGLE VALUE for every point in its domain. Your graph intersects many vertical lines more than once, so it can't be the graph of a (single-valued, by definition!) function. Also, if the domain were to extend outside where your graph projects onto the horizontal axis, you would also have points in the domain with NO value assigned (at least not shown on the graph), another violation of the definition of "function."