Let 𝑔(𝑥) g ( x ) be the inverse of 𝑓(𝑥)=𝑥^3+5𝑥+7 f ( x ) = x 3 + 5 x + 7 . Calculate 𝑔(13) and g’(13)

Question

Let g(x) be the inverse of f(x)=x^3+5x+7. Calculate 𝑔(13) [ without finding a formula for 𝑔(𝑥)g(x) ] and then calculate 𝑔′(13) using the formula 

g’(x) = 1 / f’(g(x))

g(13)= ?

g’(13)=?

Answer 1

f(x) = x³ + 5x + 7 So, f'(x) = 3x² + 5

f(1) = 13, so g(13) = 1

g'(13) = 1 / f'(g(13)) = 1 / f'(1) = 1 / 8

The slope of the tangent line to the graph of y = f(x) at the point (1,13) is f'(1) = 8

So, the slope of the tangent line to the graph of y = f^-1(x) = g(x) at the point (13,1) is g'(1) = 1 / 8