Chapter 1 Lesson F: The Reciprocal Function

Reciprocal function: is defined as x---> 1/x or f(x)= 1/x

Properties of f(x)= 1/x undefined at x=0

• graph exists in 1st and 3rd quadrant
• it is symmetric about the y=x and y=-x
• as x ---> infinite ; f(x)----> 0 ( as it is approaching infinite the value is getting closer to 0)
• as x ---> 0; f(x)---> infinite
• as x ---> (negative) infinite: f(x)

     It is asymptotic to the x-axis and to the y-axis the graph gets closer to the axis as it gets further from the line orgin.

Chapter 1 lesson E: Sign Diagrams

Sometimes we need to know when the function is positive, negative, zero or undefined without the whole picture = sign diagra

Sign Diagrams consist of :

• horizontal line (the x- axis)
• positive/negative signs indicating that the graph is above and below the x-axis.
• critical values: the numbers written below the line which are the graph's x-intercepts and points where is it undefined.

Examples: Drawing a sign diagram

 

sign diagram:

• a sign change occurs about a critical point for single factors such as (x+2) and (x-1). This indicates cutting of the x-axis
• no sign change occurs about the critical value for squared factors such as (x-2)^2. This indicates touching of the x-axis.
• function is undefined at x=0
• in general, when a factor has an odd power there is a change of sign about the critical value.
• when a factor has an even power there is NO change in sign about the critical value.
• example: draw a sign diagram : (x+3)(x-1) 
• let x+3 = 0 which equals x=-3
• let x-1= 0 which equals x=1

*sorry for the inconvenience but the question marks above, in the equations, are actually x.

Chapter 1 Lesson D: Compile Functions

Given f:x ---> f(x) and g:x ---> g(x) the composite function of f and g will convert x into f(g(x))

• f.g is used to represent the composite function of f and g. It means "f following g".
• (f.g) = f(g(x)) / f.g: x---> f(g(x))

Example:

• f: x ---> 2x + 1 and g: x ---> 3-4x find in simplest form.
• a. (f.g)(x) so f(x) = 2x+1 and g(x) = 3-4x
• f(g(x)) = 2(3-4x)+1 = 6-8x+1 = 7-8x
• g(f(x)) = 3-4(2x+1) = 3-8x-4     -8x-1

Chapter 1 Lesson C: Domain and Range

Domain: of a relation is the set of permissible values that x may have

Range: of a relation is the set of permissible value that y may have

Chapter 1 Lesson B: Functions

Function machines: illustrate how functions behave

F(x) is the value of y for a given value of x so y = F(x)

• y = f(x) ----> 2x^2 - 3x find f(5)
• 2(5)^2 - 3(5) = 50 - 15.  f(5)= 35
• f(x) = 5- x-x^2
• a. f(-x) = 5-(-x) - (-x)^2 = 5+x-x^2
• b. f(x+2)- (x+2)^2 --->  5-x-2-x^2 - 4x-4
• f(x+2)= -x^2 -5x-1

Chapter 1: Functions (Lesson A)

Section A: relations and Functions

Domain- the set of possible values on the horizontal axis. (x-axis)
Range- the set of possible values on the vertical axis. (y-axis)

Relations- any set of points pn the cartesian plane.
Equation- expresses the relation which connects the variable X and Y.

• y = x+3 (line equation), x = y^2 +1 (parabola)

These equations generate sets of ordered pairs.
             there are some relationships that can't be defined by an equation

Functions: a relation in which no two different ordered pairs have the same x-coordinate function is a special type of relation.

Testing for functions, algebric
             id a relation is given as an equation and the substitution of any value of x results in one, and only one, value of y, we have a function.
example-
y = x+3, let x equal any number.
x = 3, y = 3+3 =6 (correct/yes)

x = y^2+1 let x = 4.
x=4, 4 = y^2 +1 as we do the work we get negative square root of 3. (incorrect/no)

Geometric Test: vertical line test
         if we draw all possible vertical lines on the graph of a relation, the relation:
1. is a function if each line cuts the graph no more than once.
2. its not a function if the line cuts the graph more than once




Guess which graph passes the test.

Graphical Notes: if a graph contains a small open circle (end point), the endpoint is not included.
       if it contains a small filled in circle endpoint the endpoint is included.
       it it contains an arrow head at the end then the graph continue indefinitely in the general direction or the shape may repeat as it has done previously.