Inverse Function Notes

Chapter 3.7 Notes Inverse Functions Inverse Relation – interchanging the first and second coordinates of each ordered pair in a relation or interchanging the variables A graph of a function and its inverse is reflected over y = x Inverse of a function written as f -1 One – to – one function If a ≠ b, then f(a) ≠ f(b) or if f(a) = f(b), then a = b Not 1 – 1 1–1 Properties - inverse is a function - domain of f is the range of f -1 - range of f is the domain of f -1 - function is increasing over its domain or is decreasing over its domain Horizontal – Line test - fails then the function is not one – to –one and its inverse is not a function Obtain a formula for inverse - change f(x) to y - interchange x and y - solve for y - replace y with f -1(x) Inverse functions and composition

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if a function is one – to – one then

( f o f 1 )  x   x and  f 1 o f   x   x

Restricting a domain - if the inverse of a function is not a function , the domain of the function can be restricted to allow the inverse to be a function Examples 3.7 #5, 11, 17, 19, 21, 23, 29, 35, 41, 45, 51, 55

3.6 Operation of Functions If the ranges of the functions f and g are subsets of the real numbers, then 1) (f + g)(x) = f(x) + g(x) 2) (f – g)(x) = f(x) – g(x) 3) (f ⋅ g)(x) = f(x)g(x) 4) (f/g)(x) = f(x)/g(x) g(x) ≠ 0 The domain of each function, unless otherwise restricted, is the set of real numbers x that are in the domains of BOTH f and g.

The composite function f  g is defined by ( f  g )( x ) = f ( g ( x ) ) The domain of f  g consists of all those numbers in the domain of g for which g(x) is in the domain of f.

3.6 #11, 15, 19, 25, 27, 35, 39, 43, 51, 52, 59, 69, 81