Answer
(a) A function is a map from one set $X$ to another $Y$ such that every element of the set $X$ maps to exactly one element of set $Y$.
(b) The graph of the function $f$ is the set of points in the $xy$ plane that have $x$ as their first and $f(x)$ as their second coordinate: $(x,f(x))$.
(c) If every vertical line intersects the curve only once then that curve is the graph of a function.
Work Step by Step
(a) A function is a map from one set $X$ to another $Y$ such that every element of the set $X$ maps to exactly one element of set $Y$.
There can be no multiple elements of $Y$ corresponding to the same element of $x$. However, multiple elements of $X$ may be mapped to the same element of $Y$.
(b) The graph of the function $f$ is the set of points in the $xy$ plane that have $x$ as their first and $f(x)$ as their second coordinate: $(x,f(x))$.
(c) If every vertical line intersects the curve only once then that curve is the graph of a function.
The vertical line represents the constant value of $x$. This condition means that to each $x$ corresponds unique value $f(x)$ which is required by the definition of the function.
