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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.
Parallel curves
Published in International Journal of Mathematical Education in Science and Technology, 2022
Richard Dexter Sauerheber, Tony Stewart
It is possible to generate curves of functions that give the appearance of being completely parallel along their domains. Shifting an odd function that passes the horizontal line test, such as x3, along the horizontal and vertical axes demonstrates this. Functions given by (x + 1)3 + 1 and x3 are graphed in Figure 4. The curves do not intersect, as required for parallel curves, and have equal length line segments between any pair of corresponding points with the same slope. For any horizontal shift a and vertical shift b, the derivative of (x + a)3 + b at position (x − a, b) is the same as the derivative of x3 at that value of x, namely 3x2. Further, line segment lengths connecting any corresponding equal-sloped points are all of equal length, given by L = (a2 + b2)0.5. However, these line segments are not simultaneously perpendicular to both curves and are thus not parallel according to our working definition above.