The document discusses inverse functions, including: - An inverse function undoes the output of the original function by relating the input and output variables. - For a function to have an inverse, it must be one-to-one so that each output is paired with a unique input. - To find the inverse of a function, swap the input and output variables and isolate the new output variable.

1. Inverse Function

2. Today’s Outline
o Inverse function
o History behind function
o Finding Inverse of a function
o Application
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3. 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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Function
Y=f(x)
Input variables
X
Output variable
Y

4. Example
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5. 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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6. 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)) = x
f(f-1 (x)) = x
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7. 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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8. One to one Functions
Every element of the
range corresponds to
exactly one element of
the domain.
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9. Why all functions don’t have Inverse ?
All functions are not one to one functions.
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10. Why all functions don’t have Inverse ?
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A
B
C
D
1
2
3
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11. Why all functions don’t have Inverse ?
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1
2
3
4
A
B
C
D
No longer a function

12. 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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13. Problem
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14. Solution
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15. Solution
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16. 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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17. Any Query ?

18. Thank you