MA005: Calculus I

Definition of Function:

A function from a set $X$ to a set $Y$ is a rule for assigning to each element of the set $X$ a single element of the set $Y$. A function assigns a unique (exactly one) output element in the set $Y$ to each input element from the set $X$.

The rule which defines a function is often given by an equation, but it could also be given in words or graphically or by a table of values. In practice, functions are given in all of these ways, and we will use all of them in this book.

In the definition of a function, the set $X$ of all inputs is called the domain of the function. The set $Y$ of all outputs produced from these inputs is called the range of the function. Two different inputs, elements in the domain, can be assigned to the same output, an element in the range, but one input cannot lead to 2 different outputs.

Most of the time we will work with functions whose domains and ranges are real numbers, but there are other types of functions all around us. Final grades for this course is an example of a function. For each student, the instructor will assign a final grade based on some rule for evaluating that student's performance. The domain of this function consists of all students registered for the course, and the range consists of the letters $A, B, C, D, F,$ and perhaps $W$ (withdrawn). Two students can receive the same final grade, but only one grade will be assigned to each student.