Transcendental Function By Danang Mursita

The Inverse Function Let f and g are continuous function y = f(x) = f( g(y)) = (fog)(y) for all y in Dg = B x = g(y) = g( f(x) ) = (gof)(x) for all x in Df = A If it is obtained the above compositions then g is said as an inverse of f. Notation g = f -1. So, (fof -1 ) = (f -1 of) = I ( identity function) B   A x y f g f(x) = y and g(y) = x

The Inverse Function How to find the inverse of y = f(x) ? 1.Substitute x by y, so that x = f(y) 2.Arrange this form, x = f(y) into y = g(x) 3.So, g(x) = f -1 (x) Example : find the inverse of 1.f(x) = x – 5 2.f(x) = (x – 2 )/(x + 3) 3.f(x) = x 2 + 4

The existence of Inverse If the function f(x) is increasing neither decreasing on interval I then f(x) has inverse on interval I. The function f(x) has an inverse if and only if f(x) is one – to – one function Definition : f(x) is one – to – one function – if f(x 1 )  f(x 2 ) then x 1  x 2 or – This function f(x) has only one point of intercept with the any line y = b

The Properties of the function and its Inverse The domain of f(x) equal with the range of f -1 (x) and the range of f(x) equal with the domain of f -1 (x) The graph of f(x) and f -1 (x) are symmetry by the line y = x. y = x

Problems Find the inverse of these functions

The logarithm and the Exponential Function The Logarithm function is inverse of the exponential function or in the contrary. Notation : – y = b log x  x = b y with x and b are real positives

The Properties of logarithm b log b = b log 1 = b log (ac) = b log (a/c) = b log (a c ) =

The Natural Logarithm Function Notation and Definition : – e log x = ln x with natural number e = 2,718… – Differentiation and integration: – f ‘(x) = 1/x  – if f(x) = ln u then f ‘(u) = u’/u 

Problems Find dy/dx 1.y = ln(2x) 2.y = ln (sin x) 3.y = ln ( ln x ) 4.y = x 2 ln x Evaluate

The Natural Exponential Function If y = e x = exp(x) then x = ln y The properties of exponential function – ln e x = x for all x – e lny = y for all y > 0 Differentiation and integration – f(x) = e u  f ‘(u) = e u u’ –

The General Exponential Function Notation – f(x) = a x = e lna x = e xlna – f(x) = a u = e lna u = e ulna Differentiation – Integration –

The General Logarithm Function Notation – f(x) = b log x = e log x/ e log b = ln x / ln b – f(x) = b log u = e log u / e log b = ln u / ln b Differentiation –

Problems Find dy/dxEvaluate

The Inverse Trigonometric Function Trigonometric function, f(x) = sin x is not one-to-one function, so it has not an inverse function If it has an inverse function, then its domain will be bounded on some interval 1 X Y

The arcs Sinus Domain of Sinus Function, f(x) = sin x Notation Relation between the function and its inverse

The Arc Sinus Differentiation, y = sin -1 u Integration y u 1

The others of Inverse Trigonometric

Problems Find dy/dxEvaluate

The Hyperbolic Function Sinus Hyperbolic and Cosinus hyperbolic

Differentiation and Integration of Hyperbolic Function

Problems Find dy/dxEvaluate

The Power of Function Let y = f(x) g(x). Then ln y = g(x) ln f(x) The derivative, dy/dx : Example : find dy/dx

Problems : Find dy/dx

The Indeterminate Form of Limit Let y = f(x) g(x). Then lim y = lim f(x) g(x) has indeterminate form : 0 0,  0 and 1 . How to solve this limit ? – ln lim y = ln lim f(x) g(x)  lim ln y = lim g(x) ln f(x) – Let lim ln y = A. Then lim y = e A. Examples :

Problems