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...

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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.

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`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. 

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