Functions and Continuity
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Inverse of a Function: monotone increasing or decreasing functions
Equations are omitted for technical reasons - download the original pdf
If a function f maps x to y, then the inverse of that function, written [symbol], maps y to x. The inverse of a function reverses the process represented by that function. However, not all functions have inverses. This is because functions can be ''many-one'' or ''one-one''. An example of a many-one function is [Equation goes here - download the original pdf to see it.] [Diagram goes here - download the original pdf to see it.] This is many-one because there are two values in the domain giving the same value in the co-domain: [equation]. A many-one function is a function such that there are two or more arguments in the domain giving the same value in the co-domain. A many-one function cannot have an inverse because the arguments of the ''inverse'' would have more than one value, and a function must specify just on e value for each argument. A one-one function specifies for each argument just one value. For a function to have an inverse it must be one-one. A one-one function is either always increasing or always decreasing. An always-increasing function is also called a monotone increasing function, and an always-decreasing function is also called a monotone decreasing function. To prove that a function is monotone increasing or decreasing we use analytic methods; that is, it is an application of the differential calculus. For a function with out points of inflexion to be monotone increasing then its derivative is always positive. [Equation goes here - download the original pdf to see it.] As the definition indicates we will count a function as monotone increasing if it is increasing throughout the domain except, perhaps, where it has a point of inflexion. [Diagram goes here - download the original pdf to see it.] Example (5) [Example goes here - download the original pdf to see it.] A one-one function can be created from a many-one function by restricting the domain. This means diminishing the size of the domain so that the function becomes increasing or decreasing on the reduced domain. For example [Diagram goes here - download the original pdf to see it.] [Equation goes here - download the original pdf to see it.] The domain is restricted to contain only positive real numbers. The inverse of this restricted function [equation]is called the square root.[Equation goes here - download the original pdf to see it.]
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Contents of
Functions and Continuity
1 Functions
2 Graph
3 Inverse of a Function: monotone increasing or decreasing functions
4 The reciprocal function - singularities
5 Functions defined piecewise on their domain
6 Limits
7 Formal definition of a limit
8 Formal definition of the limit of a function at a point x = a
9 Informal arguments
10 Continuity
11 Combining limits
12 Quotients
13 Image set
14 Inverse image
15 Odd and even functions
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