# Find the inverse function, and state the domain and range of the inverse function : y=8(x-1)^3

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You need to find the equation of inverse function, hence, you should write x in terms of y such that:

`y = 8(x-1)^3 => y/8 = (x-1)^3`

You need to raise to the power `1/3` both sides, to remove the cube of binomial `x -1` , such that:

`(y/8)^(1/3) = ((x-1)^3)^(1/3)`

Converting the power into the cube root yields:

`root(3)(y/8) = x - 1 => (root(3)y)/2 = x - 1`

You need to isolate x to the right side such that:

`(root(3)y)/2 + 1= x`

Using the standard notation yields `y = ((root(3)x) + 2)/2` .

Hence, evaluating the equation of the inverse function yields `f^(-1)(x) =((root(3)x) + 2)/2.`

Notice that there is no restriction for domain of the inverse function since the cube root accepts positive and negative radicands.

**Hence, evaluating the domain and the range of the inverse function `f^(-1)(x) = ((root(3)x) + 2)/2` yields that both are represented by the set of real numbers.**

`y = 8(x-1)^3`

To determine the inverse function, interchange x and y.

`x = 8(y-1)^3`

Then, solve for y. To do so, divide both sides by 8.

`x/8 = (8(y-1)^3)/8`

`x/8=(y-1)^3`

Then, take the cube root of both sides.

`root(3)(x/8)=root(3)((y-1)^3)`

`root(3)(x)/2 = y-1`

And add btoh sides by 1.

`root(3)(x)/2 + 1 = y -1+1`

`root(3)(x)/2 + 1 = y`

Then, replace y with `f^(-1)(x)` to indicate that it the inverse function of `y=8(x-1)^3` .

**Hence, the inverse of the given function is `f^(-1)(x) = root(3)(x)/2 + 1` .**

**Using the properties of a cube root function `root(3)(x)` , the domain of the inverse function is all real numbers and its range is all real numbers too. **