Question

`\sqrt{ - x ^(3) }`

what would this be? The cool equation is Y=2-X^3. I'm doing Inverse Functions.

Answer 1

I'm guessing you're given the function
`y(x)=2-x^3`, and you're asked to find the inverse function
`y^(-1)(x)`. To do this, swap
`x` and
`y`, then solve for
`y`:

`x=2-y^3\implies y^3=2-x\implies y=(2-x)^(1/3)=\sqrt[3]{2-x}`

so that the inverse function is

`y^(-1)(x)=\sqrt[3]{2-x}`

Just to verify:

`y(y^(-1)(x))=y(\sqrt[3]{2-x})=2-(\sqrt[3]{2-x})^3=2-(2-x)=x`

`y^(-1)(y(x))=y^(-1)(2-x^3)=\sqrt[3]{2-(2-x^3)}=\sqrt[3]{x^3}=x`

But in case you're actually only interested in computing the square root, first we note that
`\sqrt x` (the real-valued square root) is only defined as long as
`x\ge0`. So
`√(-x^3)` is defined as long as
`-x^3\ge0`, or
`x^3\le0`, or equivalently
`x\le0`. Under this condition, we could write

`√(-x^3)=√(-x* x^2)=√(-x)√(x^2)`

We can simplify this further, but we have to be careful. Suppose
`x=-1`. Then
`x^2=(-1)^2=1`. But we get the same result if
`x=1`, since
`x^2=1^2=1`. There are two possible values of
`x` that given the same value of
`x^2`, so to capture both of them, we take
`√(x^2)=|x|`, the absolute value of
`x`. Then

`√(-x^3)=|x|√(-x)`

We can't simplify the square root term further than this.