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.

Inequalities

INEQUALITIES

Inequality- Compares two numbers or expressions.
Solving an inequality is similar to the process of solving an equation, but there are a few exceptions.
Such as: When both sides of an inequality are divided or multiplied by a negative number, the direction of the inequality symbol must be reversed.

Properties of Inequalities

1. Transitive Property

      

2. Addition of Inequalities

    

3. Addition of a Constant

  

4. Multiplying by a Constant

 ,

 , 

A.  Linear Inequalities

              add 7 to each side

              subtract 3x from each side

                        combine like terms

                             divide each side by 2

                            Solution set consists of all real numbers that are greater than 8.

B.  Polynomial Inequalities

                                            

                               factor

                               Treat as equation

                  Split

                       Solve.     Critical Values: X-intercepts. Only places where the function could change  from 

being negative to positive or vice versa.

Number Line

Test Intervals:       

Pick a test value for each test interval. Plug value into original inequality to see if answer is negative or positive.

Test Interval:

Test Value:  

Result: 

Test Interval:  

Test Value: 

Result: 

Test Interval: 

Test Value: 

Result: 

               SOLUTION

                            INTERVAL NOTATION

Graphing can verify your solution. If we graphed this inequality it would look like the parabola below. We can see from the graph that the interval of -2 to 3 is below the x-axis and is thus less than 0. 

A shortcut:

1.  Solve for the x-intercepts, places where the function changes from being negative to positive and 

vice versa :   -2, 3

2. Examine the leading coefficient of your original quadratic function:  +1 implies parabola opens upward

3. Sketch a graph of the function

:

4. Examine Inequality and Graph:   Original inequality stated that solutions were < 0.     On the graph the negative solutions are in the interval   . So the solution can be any real number within the interval of .

If you would like another example of solving a polynomial inequality, here's a neat video. 

http://www.youtube.com/watch?v=4lE3PXm9uWQ

Hope this was helpful!

-Leah