Chapter 1 Functions

Chapter 1

Function: An mathematical relation is a function if and only if there is one 'y' output to each 'x' input

Vertical Line Test

A graph is a funtion if no vertical line drawn will intersect the graph at more than one point.

Red Flags (not a function if...):

1. y is squared  (if you have a vertical parabola)
     
ie:
                                    
 This graph does not pass the vertical line test.
This is because both y and its negative will have the same x value.  That is two y values for one x.  




      

               

2.  
  

This graph also does not pass the vertical line test.
                                                  This is the same reasoning as for when y is squared





3. You have two coordinates with the same x-value
     ie: (-2, 3) , (-1, 6) , (0, 7) , (-1, 5)

Evaluating in Function Notation

Given f(x) = x² + 2x – 1, find f(2)
f(2) = (2)² +2(2) – 1                    Plug in 2 for x       = 4 + 4 – 1                         Evaluate       = 7



Given f(x) = 2x² + x, find f(x+2)
         Plug in (x+2) for x
                Evaluate

 
=(x+2)(2x+5)





Piecewise Functions


Evaluate f(x) when (a) x = 1 (b) x = 0 and (c) x = -1

You pick a relation to use depending on which condition x satisfies.  In this equation, for example, 0 and  -1 both satisfy x<1 so you evaluate with .
On the other hand, 1 satisfies    so you evaluate with x+2

Solution for the given Piecewise Functions examples:

(a) x = 1,

f(x) = x + 2

f(1) = 1 + 2 = 3


(b) x = 0,

f(x) = x² - 1

f(0) = 0² - 1 = -1


(c ) x = -1

f(x) = x² - 1

f(-1) = (-1)² - 1= 0




More on function notation: http://www.youtube.com/watch?v=Kj3Aqov52TY

-Hannah S.