how do i veriify it has an inverse?

Mathematics
Tutor: None Selected Time limit: 1 Day

Oct 6th, 2015

Thank you for the opportunity to help you with your question!

a) First we find the inverse of the function by switching x and y, then solve for y:

x = y+10 / y - 2  

Cross multiply to get

x(y - 2) = y+10

xy - 2x = y+10

xy - y = 2x+10

y(x - 1) = 2x+10

y = 2x+10 / x - 1

So, the inverse f^-1(x) = 2x+10 / x - 1

b) f^-1(7) = 2(7)+10 / 7 -1 = 24/6 = 4

c) (f^-1)'(x) = (x - 1)(2) - (2x+10)(1) / (x - 1)^2

= 2x - 2 - 2x - 10 / (x - 1)^2

= -12 / (x - 1)^2

(f^-1)'(7) = -12 / (7 - 1)^2 = -12 / 6^2 = -12/36 = -1/3

Please let me know if you need any clarification. I'm always happy to answer your questions.
Oct 6th, 2015

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