Given f(x) and g(x)=f^-1(x).

Question

If f(1)=4 and f'(1)=-3, then find g'(4).

...No idea where to start with this. Please help?

Thanks much!

Answer 1

you don't state what type of function f(x) is, but since only 2 bits of information are given about it, let's assume it is linear, or else we would need more data

let f(x) = ax + b , where a and b are constants
f'(x) = a
but we are told f'(1) = 3
since a is a constant and f'(x) = a
a = -3
also f(1) = 4
a + b = 4
-3 + b = 4
b = 7
so f(x) = -3x + 7

then g(x) = (x-7)/-3 or -x/3 + 7/3
( I did assume you know how to take the inverse of a linear function)

g'(x) = -1/3 , independent of the value of x
thus g'(4) = -1/3

Answer 2

Ok, I have the idea, but there's one thing that's really bugging me that I don't get...

Why would a = -3?