Assuming that y is a function of x, that follows from the definition of a function. For each x-value, there can only be one y-value. The definition of a function is that (in this case), for every value of "x", a value of "y" can be calculated unambiguously. In the more general case, for every combination of the independent variables, a single value for the dependent variable can be calculated unambiguously.
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explain why a function has at most one y-intercept
There is no slope nor intercept because there is no equation, simply an expression.
A graph is represents a function if for every value x, there is at most one value of y = f(x).
One to one functions on a graph can vary. To determine if a function is one to one, a horizontal line can only intersect the function once. If it intersects the function more than once, it is not a one to one function.
For a 2-dimensional graph if there is any value of x for which there are more than one values of the graph, then it is not a function. Equivalently, any vertical line can intersect the a function at most once.