# jaque mate

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Functions, equations, and inequalities

All one to one functions have inverses which are functions and this means that the function is reversible.

A function g is a one to one function if and only if for all u and v in the domain of g, g(u) = g(v) -> u = v

f and g are inverse functions, written f = g-1 or g = f-1, if and only if f ° g(x) = x for all x in the domain of g and g ° f(x) = x for all x in the domain of f.

g ° f(x) = g(f(x))Whose domain is the set of all x in the domain of f,for which f(x) is in the domain of g.

(f+g)(x) = f(x) + g(x)(f-g)(x) = f(x) - g(x)(f*g)(x) = f(x) * g(x)(f/g)(x) = f(x) / g(x)

Definitions of Basic Arithmetic Operations on Functions

Defintion of g ° f

Inverses of Functions

The logic of equation solving

let f, g, and h be functions. then, for all x in the intersection of the domains of functions f , g and h,1. f(x) = g(x) <=> f(x) + h(x) = g(x) + h(x)2. f(x) = g(x) <=> f(x) ° h(x) = g(x) ° h(x)3. if h is one to one, then for all x in the domains of f and g for which f(x) and g(x) are the domain of h, f(x) = g(x) <=> h(f(x)) = h(g(x))

- Only 1-1 functions can undergo reversible changes.- Functions to an even power are not 1- 1 functions, therefore they cannot be reversed.- For example x elevated to 2 is not a 1-1 function because there are two values of x for each of y. besides its inverse is not even a function.-And one way to know if a function is 1-1 is noticing that there is just one value of y for each of x and one value of x for each of y.

attention!