How do you find ? a) the inverse g(x) of the function f(x)? b) graph f and g together? c) verify the derivative formula for f and g at the point c=2? {If f=x^3-1}

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Answer 1 · 2018-05-23T12:25:03

Part (A):

We seek the inverse:

# g(x) = f^(-1)(x)# where #f(x)=x^3-1#

Let #y=f^(-1)(x) iff x=f(y)#, Then:

# x=y^3-1 #

And now we rearrange:

# y^3 = x+1 #

# :. y = root(3)(x+1) #

Hence, we have:

# g(x) = f^(-1)(x) = root(3)(x+1) #

Part (B):

The graphs of the functions are a reflection in the line #y=x#
graph{(y-(x^3-1))(y-(x+1)^(1/3))(y-x)=0 [-10, 10, -5, 5]}

Part (C):

The question is unclear as to what we should verify the derivatives with, but the derivative are given by:

# f'(x) = 3x^2 #

# g'(x) = 1/3(x+1)^(-2/3) #

Thus when #x=2#, we have:

# f'(2) = 12 # and # g'(2) = root(3)(3)/9 #