For each function, find a domain on which the function is one-to-one and non-decreasing, then find an inverse of the function on this domain.

Question

For the function given below, find a domain on which the function is one-to-one and non-decreasing, then find an inverse of the function on this domain.

f(x)=4-2x^3

Answer 1

If you graph this function:

you can see that although it is one-to-one, it is always decreasing (never non-decreasing). Based on that it sounds like that's all you need to say.

The function does however have an inverse. To find it start with replacing "f(x)" with "y":

y = 4 - 2x³

Then swap the places of "x" and "y":

x = 4 - 2y³

The solve for "y":

2y³ + x = 4

2y³ = 4 - x

y³ = (4 - x)/2

y = cuberoot((4 - x)/2) or y = ³√[(4 - x)/2]

Then change "y" to f^-1(x):

f^-1(x) = ³√[(4 - x)/2]