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F(x)= (2e^x-7)/(7e^x+6) f^-1(x)=?

F(x)= (2e^x-7)/(7e^x+6) f^-1(x)=?

Grade:12th pass

2 Answers

Vikas TU
14149 Points
9 years ago
f(x) = (2e^x - 7)/(7e^x + 6) 

f(x) = {2(e^x - 7/2)}/{7(e^x + 6/7)} 

f(x) = (2/7)(e^x - 7/2)/(e^x + 6/7) 

Let x = f^-1(x) 

f(f^-1(x)) = (2/7)(e^f^-1(x) - 7/2)/(e^f-1(x) + 6/7) 

The left side becomes x by definition: 

x = (2/7)(e^f^-1(x) - 7/2)/(e^f^-1(x) + 6/7) 

Multiply both sides by 7/2: 

7x/2 = (e^f^-1(x) - 7/2)/(e^f^-1(x) + 6/7) 

Add zero to the numerator in the from of + 6/7 - 6/7
 
continue...................
Vikas TU
14149 Points
9 years ago
7x/2 = (e^f^-1(x) + 6/7 - 6/7 - 7/2)/(e^f^-1(x) + 6/7) 

7x/2 = 1 - (6/7 + 7/2)/(e^f^-1(x) + 6/7) 

Multiply by 2: 

7x = 2 - (12/7 + 7)/(e^f^-1(x) + 6/7) 

Multiply numerator and denominator by 7/7: 

7x = 2 - (12 + 49)/(7e^f^-1(x) + 6) 

7x - 2 = - 61/(7e^f^-1(x) + 6) 

7e^f^-1(x) + 6 = 61/(7x - 2) 

7e^f^-1(x) = 61/(7x - 2) - 6 

e^f^-1(x) = 61/(49x - 14) - 6/7 


f^-1(x) = ln|61/(49x - 14) - 6/7|

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