If f(g(x)) = x and g(f(x)) = x, and f(x) passes through the point (5,-7), and g'(-7) = -2/3, then find the value of f'(5).

So it is given that f(x) and g(x) are inverse of each other.

Then f(5) = -7 and hence g(-7) = 5

Now let's use f( g( x ) ) = x

Now take deviation. Use chain rule:

f'( g(x) ) g'( x ) = 1

Plug in x = -7

f'( g( -7 ) ) g'( -7 ) = 1

f'( g( -7 ) ) · -2/3 = 1

f'( g( -7 ) ) = -3/2

f'( 5 ) = -3/2

Hope this helps.