December 6: Introduction to Functions

Section 1.1 is an introduction to functions. A function, according to our textbook is "... a set A to a set B is a relation that assigns to each element x in the set A exactly one element y in the set B. The set A is the domain (or set of inputs) of the function f, and the set B contains the range (or set of outputs).

Simplified, this means that a function is a relationship between two things in which for every x value (the input) there is only one y value (the output).

For example, the relationship between the side length of a square and its area is a function. The reasoning behind this is that every square that has the same side length will have the same area. (The input, in this case the side length will always have the same single output, in this case- the area)

The relationship between the height of a rectangle and the area of a rectangle, though, is not a function. In this example, the area depends upon the width of the rectangle for the area, not just the height. Let’s say that one rectangle’s height is 3 and the width is 5, then that rectangle’s area is 15. If the second rectangle’s height is still 3 but it has a width of 7, then its area will be 21 not 15. That means that the input (3), has more than just one output.

However, a function is still a function if an output has more than one input. For example, the set of points: {(-5,2), (-2, 1), (0,0), (3,7), (4,2)} is a function even though the -5, and 4 both result in 2.

A set of points that is not a function is: {(-5,2), (-2, 1), (0,0), (3,7), (4,2), (3,-3)} because one element in the domain, 3, corresponds with more than one range 7 & -3.

Today, we also talked about Function Notation or f (x)

An extremely important lesson that we were taught about function notation is this: y = f (x) ---- (x, y). This means that the point (x, y) is on the graph of f.

For example:

f (x)= 2x +1

Then...

f (2)=2(2)+ 1=5

and

f (a)= 2(a) +1

and

f (!)=2(!) +1

* All these examples are assuming that y (the output) is a function of x (the input).