Undoing it                                                is solving                                            and requires new numbers
Addition Subtraction x+2=6: x=6-2=4, 
x+6=2: x=2-6=-4
Integers
Negative numbers -4=2-6=3-7=...
Functions: f(x)=x+c shifts Expand addition to Integers Geometry: Shifts (Horizontal/Vertical)

 

Doing  it again                                           Undoing it                                                is solving                                            and requires new numbers
Multiplication Division x*2=6: x=6/2=3, 
x*6=2: x=2/6=1/3=3^{-1}
Rational numbers 1/3=2/6=3/9=...
Multiplication and Addition
a(x+y)=ax+ay (Distributive Law)
Functions: f(x)=cx scalings, linear, polynomial
 
Expand addition, multiplication to
Rational numbers
Geometry: Scalings (Line, Plane)

 

Doing it again                                            Undoing it                                                is solving                                            and requires new numbers
Powers (Squaring, Cubing, etc.) x^2,x^3 Roots (Square root, Cube root, etc.)
x^{1/2},x^{1/3}
x^2=4: x=4^{1/2}=2, 
x^2=2: x=2^{1/2}={\sqrt 2}
Certain irrational and complex numbers
Powers and Addition
(x+y)^2=x^2+2xy+y^2, Binomial

Powers and Multiplication
(xy)^n = x^n y^n

Functions: Expand addition, multiplication, powers to Irrational and Complex numbers Geometry (Scalings, Rotations)

 

Doing it again                                            Undoing it                                                is solving                                            and requires new numbers
Exponentials 
2^x, 3^x
Logarithms
log_2 (y), log_3(y)
2^x=8: x=3=log_2 (8)
2^x=3: x=log_2 (3)
Certain irrational numbers
Exponentials and Addition
a^{x+y}=a^x a^y

Exponentials and Multiplication
a^{xy}=a^x a^y

Functions: Expand addition, multiplication, exponentials to Irrational and Complex numbers Geometry

 
 
 
 
 
 

The pattern of Precalculus

Iteration (do it again)   Inversion (Undo it/Solve)  Expansion   Interaction  Kinds of numbers, Kinds of functions (linear,polynomial,rational,transcendental)  Geometry

Addition (Natural Numbers)                                           Subtraction (Integers)
                                                                                   Solving x+2=6 and x+6=2
 
 
 
 

 Multiplication (Natural Numbers/Integers)                             Division (Rational Numbers)
Multiplication with Addition (Distributive Law) a(x+y)=ax+ay
 
 
 
 
 

          Powers/Exponentials (Natural Numbers, Integers, Rational Number)        Roots/Logarithms (Irrational numbers, Imaginary/Complex Numbers)

Power/Addition: Pascal/Binomial (x+y)^n= \sum_{j=0}^n ...
Power/Multiplication  (xy)^n=x^n y^n
Exponential/Addition  a^{x+y}=a^x a^y
Exponential/Multiplication a^{xy}={{a^x}^y}

And Calculus

Limit/Addition-Subtraction
Limit/Multiplication-Division
Limit/Iteration-Composition-Inversion

Derivative-Integral
Higher Derivative Multiple Integral

Derivative/Integral / Addition/Subtraction, Multiplication/Division, Iteration/Composition/Inversion, Power,Exponential,Circular functions

Connection to geometry, Connection to infinity