Inverse functions

Inverse Function

Today’s Outline

o Inverse function

o History behind function

o Finding Inverse of a function

o Application

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What is Function ?

A function is a mathematical process that uniquely relates the value of one variable to the value of one or more other variables.

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FunctionY=f(x)

Input variables

XOutput variable

Y

Example

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Historical Background

Galileo gave the statements of dependency of one on another.

In 1673 Leibnitz used the word ‘function’.

In 1734 the notation f(x) was introduced by Euler.

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What is Inverse Function ?

Let f be a function with domain D and range E. The inverse of f is the function f-1 defined by:

f-1 (b) = a where a is chosen so that f(a)=b

So, f-1 (f(x)) = xf(f-1 (x)) = x

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What functions are Invertible ?

In order for f-1 to be a function there must be only one a in D corresponding to each b in E.

Such functions are called one to one. The graph of such functions passes horizontal line test. If f is continuous, then f-1 is continuous too.

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One to one Functions

Every element of the

range corresponds to

exactly one element

of the domain. 

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Why all functions don’t have Inverse ?

All functions are not one to one functions.

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Why all functions don’t have Inverse ?

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A

B

C

D

1

2

3

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Why all functions don’t have Inverse ?

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1

2

3

4

A

B

C

D

No longer a function

Steps for Finding Inverse

Stick y for f(x)

Switch x and y

Solve for y

Replace y with f-

1

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Problem13

Solution

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Solution

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Application

Let  f  be the function that converts a temperature in degrees Celsius to a temperature in degrees Fahrenheit.

F = f(c) = 9/5 * C + 32

C = f-1 (F) = 5/9 * (F-32)

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Any Query ?

Thank you