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

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/Multiplication  (xy)^n=x^n y^n
Exponential/Multiplication a^{xy}={{a^x}^y}

And Calculus

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