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Transcript
Chapter 4
Functions and Relations
Notes
Relation – Any set that can be written as an ordered pair (x, y)
Domain - All of the x values of a relation
Range – All of the y values of a relation that correspond to the domain
Finite Set – Fixed number of ordered pairs
Infinite Set – A set that has a never ending set of ordered pairs
**** If no set or domain is not specified, the domain would always be
the real numbers!
FUNCTION - A relation that has one and only one y value for every x
value in the domain.
Functions - 1. A set is a function if it has no repeating x values
2. A Graph is a function if it passes the vertical line test
3. Even though a graph may not be a function, you can
Restrict the domain of the graph to make it a
function
Function Notation - y = , F(x), F:, F={(x,y)|,
Types of function - 1. Linear Functions - y = mx + b, Domain and Range
Is always real numbers except x = or y = lines
2. Quadratic Functions - y  ax  bx  c , Domain
Is always the Real numbers and the range depends
on the turning point.
2
3. Polynomial Functions - y  ax  .........
Where n is greater than 2. Domain is always
The real number and range depends on the power
of n.
n
4. Absolute Value Function – y = |x| , Domain is
All on the real numbers and the range is the
Positive real numbers.
5. Square Root Functions - y  x , Domain is the
Positive real numbers and the range is also the
Positive real numbers
6. Rational Functions – y 
x3
, Domain depends
x4
On the denominator and the range follows.
7. Rational Functions with a square root combination of 5 and 6.
8. Identity Function – y=x
Direct Variation - The variables are directly proportional to each other.
Function Transformations 1. Addition/subtraction of a Constant - Moves function up and down
2. Multiplying a function by a constant –
a. Constant greater than 1 - function becomes steeper
b. Constant less than 1 – slope of function decreases
3. Absolute Value of a function - The negative part of the function
reflects over the x-axis.
4. Add or Subtract a constant in the function - Moves the function
right or left.
One to one Functions - Has one y for every x and one x for every y.
Example: lines, square roots, some polynomial functions
Composite Functions - One Function inside of the other function.
Symbol - ( f  g )( x)  f ( g ( x))
Inverse Functions – If you do the composite of two functions and it comes
out to x.
Simple definition: Switch your x and y and resolve for
inverse.
Limiting domain – To find the inverse of functions that are not one to one
the domain must be limited to create an inverse
that is a function.
Circles Basic formula – Center at (0,0)
x 2  y 2  r 2 , where r = the radius
Center – Radius Formula –
 x  h   y  k 
2
2
 r 2 , where (h,k) is the Center
and r = radius of the circle.
Standard Form of Circles –
x2  y 2  dx  ey  f  0 , Basically the
Center-Radius form multipled out
Changing from the standard form to the Center radius form:
Use Completing the Square twice to change the form.
Changing from the Center radius form to the standard form:
FOIL out the binomials and move all terms to one side.
Inverse Variation - Inversely proportional.
xy  c , where c is a constant
The equation is a Hyperbola.