find the inverse of f(x)=2(x+3)^3

Question

answer key has (-6+cube root of (4x))/2

where does the 4 come from?

Answer 1

f(x) =y=2(x+3)^3

f^-1(x)= -3+ (x/2)^(1/3)

switch x and y and solve for the new y

x=2(y+3)^3

x/2 = (y+3)^3

y+3= (x2)^(1/3)

new y = f^-1(x)= -3 +(x/2)^(1/3)=-3 plus cube root of x/2

=(-6+(4x)^(1/3))/2 when you rationalize the denominator inside the cube root

Answer 2

desmos.com/calculator/qqmktgfmvx

The above graph has a sliding point on the original function and shows the segment joining that point and its image on the inverse. The midpoint of that segment lies on the line y = x.