Chapter 1: Functions and Their Graphs

1.1 - Functions

Today in class we discussed the basics of functions.

The direct definition from the book of a function is: A function f from 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).

In short, a function has exactly one output (y) for every input (x). 

Quick reminder: Domain refers to x values and range refers to y values.

Example:

Domain: 

Since x is the denominator, 

it can be any nonzero number. 

Therefore the domain, the x values, 

can be anything except for zero.

Example 2:

Domain: 

Since x is underneath the radical, 

it can only be an nonnegative number. 

Therefore the domain can only be 

greater than or equal to zero.

Functions in Real Life situations:

x (independent variable)	y (dependent variable)	Function?	Explanation
Amount of time spent studying for a test	Grade earned on test	No	Two people could study the same amount of time for the test, but ultimately receive different grades.
Length of a side of a square	Area of the square	Yes	As the side of a square changes, so does the area. For each side length, there’s only one possible area.

How to tell if it's a function: 

	Function	Not a Function
Equation	y = 7x - 8                                       (This would be a function because for each x value you put in, you’ll end up with a single y value)	y2 = 7x- 8                              (This would not be a function, because when you square each side, you’ll get a +/- value for the y, meaning two y values for an x value)
Points	(1,2) (4,3) (7,1) (8,3) (3,2)   (This would be a function as there aren’t repeats for the x value)	(1,2) (4,3) (7,1) (8,3) (3,2) (4,9) (This would not be a function since there are two points with 4 as an x value)
Graph		The vertical line test can be used to determine if a graph is a function or not. Place a vertical line (imaginary or tangible) on the graph. If it touches more than one part of the graph at once at any point, it means it’s not a function due to the multiple x values for one y value.

IMPORTANT:

F(x) = y implies that (x,y) is a point on the graph of F