Question

How do you find the range of `f(x)= (x^3+1)^-1`?

Answer 1

`{y|y!=0}`

Explanation:

Do find the range, get the domain of the function's inverse.

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

Flip `x` and `y`.

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

Now isolate `y`.

`x=(y^3+1)^-1`
`x=1/(y^3+1)`
`(y^3+1)x=(y^3+1)1/(y^3+1)`
`(y^3+1)x=1`
`y^3x+x=1`
`y^3x+x-x=1-x`
`y^3x=1-x`
`(1/x)y^3x=(1/x)(1-x)`
`y^3=(1-x)/x`
`root(3)(y^3)=root(3)((1-x)/x)`
`y=root(3)((1-x)/x)`
`color(blue)(f^-1(x)=root(3)((1-x)/x))`

Now looking for the domain of the inverse function, we will find that it will only be undefined at `x=0`. So its domain is `{x|x!=0}`.

Therefore, the range of `f(x)=(x^3+1)^-1` is `{y|y!=0}`.