The directions say "find a formula for f^-1 and verify that (f of f^-1)(x)=(f^1 of f)(x)=x. The problem is f(x)=100/(1+2^-x). Can anybody help me??
My preferred method is to set f(x) = y and solve for x in terms of y:
Take the reciprocal:
Multiply both sides by 100:
Subtract 1 from both sides:
It should be relatively easy to solve for x using logarithms. You can pick whichever base you like although I prefer base e (the natural log)
Using logarithms to get this is the best method of solving exponential equations like this, perhaps you should revise them when you next get a chance.
Since the book wants to use base 2 lets go with that:
By our log laws we know that so we can rewrite the LHS as . Do you understand how I used the laws to get to -x ?
In the equation as whole we have, after dividing by -1:
Now that we have x in terms of y rewrite x as and y as :
.
This is the same as your book's answer, just written in a different way. If you want the book's answer use the log power laws on that -1 outside the logarithm and the effect of raising a fraction to power -1