Composite Functions

Question

Let function f: R→R and function g: R→R be two functions for which f(g(x))=x and g(f(x))=x. Explain how the following is true: if f(a) is equal to b and f'(a) is equal to m (where m is nonzero), then it follows that g'(b) is equal to 1/m

Answer 1

f and g are inverse functions

f(x)= g^-1(x)

g(x)= f^-1(x)

m and 1/m are inverses

m^-1= 1/m

for example

let f(x) = 3x+2

g(x) = (x-2)/3

f(g(x)) = 3(x-2)/3+2 =x

g(f(x))= (3x+2-2)/3 = x

f'(x) = 3

g'(x)=1/3

3 and 1/3 are inverses

3^-1= 1/3

Answer 2

g(f(x)) = x

By the Chain Rule, g'(f(x))f'(x) = 1

When x = a, g'(f(a))f'(a) = 1

Since f(a) = b and f'(a) = m, g'(b)(m) = 1

So, g'(b) = 1/m.