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A Review of Calculus
Published in Richard L. Shell, Ernest L. Hall, Handbook of Industrial Automation, 2000
Let f, F be two given function with domains, Dom(f), Dom(F), and ranges, Ran(f), Ran(F). We say that f (resp. F) is the inverse function of F (resp. f) if both their compositions give the identity function, that is, if (f ∘F)(x) = (F∘f)(x) = x [and, as is usual, Dom(f) = Ran(F) and Dom(F) = Ran(f)]. Sometimes this relation is written as (f ∘ f−1)(x) = (f−1 ∘ f) (x)= x. For instance, the functions f, F defined by the rules f(x) = x2 and F(x) = x are inverses of one another because their composition is the identity function. In order that two functions f, F be inverses of one another it is necessary that each function be one-to-one on their respective domains. This means that the only solution of the equation f(x) = f(y) [resp. F(x) = F(y)] is the solution x = y, whenever x, y are in Dom(f), [resp. Dom(F)]. The simplest geometrical test for deciding whether a given function is one-to-one is the so-called horizontal line test. Basically, one looks at the graph of the given function on the xy-plane, and if every horizontal line through the range of the function intersects the graph at only one point, then the function is one-to-one and so it has an inverse function. The graph of the inverse function is obtained by reflecting the graph of the original function in the xy-plane about the line y = x.
Hyperbolic functions
Published in C.W. Evans, Engineering Mathematics, 2019
Bijections are comparatively rare, and so usually it is necessary to modify either the domain, the codomain or both in order to obtain a function which has an inverse. When this is done the inverse functions are not of course the inverses of the original functions, because the original functions are not bijections and so have no inverses. This fact is often obscured, but most people avoid the difficulty by giving these pseudo-inverse functions the name principal inverse functions.
Mathematical Preliminaries
Published in Dr Arzhang Angoshtari, Ali Gerami Matin, Finite Element Methods in Civil and Mechanical Engineering, 2020
Dr Arzhang Angoshtari, Ali Gerami Matin
If R(f)=Y, f is called onto or surjective. A function f:X→Y is called one-to-one or injective if different points do not take the same values, that is, f(x1)=f(x2), implies that x1=x2. A mapping which is injective and surjective is called bijective. A bijective function f:X→Y is invertible, that is, f admits the inverse function f-1:Y→X, where for any x∈X and y∈Y, we have f-1(f(x))=x, and f(f-1(y))=y.
The impact of lead-time uncertainty in product configuration
Published in International Journal of Production Research, 2021
Qinyu Song, Yaodong Ni, Dan A. Ralescu
A real number c is a special uncertain variable whose uncertainty distribution does not actually have an inverse function. However, for convenience, we say the real number c has an inverse uncertainty distribution .