## Calculus: Early Transcendentals (2nd Edition)

Item (i) is true: (i) For each value of x in the domain, there corresponds one unique value of y in the range. The vertical line test is used to ensure this -- a vertical line can only pass through the function graph once at each point, insuring that there is only one value of $y$ for any given value of $x$. If there were two values of $y$ for the same $x$ (for example $y=f(2)=5$ and $y=f(2)=10$), the function would not make sense -- the dependent variable $y$ must have a unique value for a given value of the independent variable $x$. Item (ii) is not necessarily true: (ii) for each value of $y$ in the range, there corresponds one unique value of $x$ in the domain. For example, if $y=f(x)=x^2$, then $f(-1)=f(+1)=1$. While (ii) can be true for some functions (e.g. an odd function, like $f(x)=x^3$), it need not necessarily be true for all functions.