Question

Find the inverse of the function, and determine if the inverse is a function.

y= cuberoot of (x-5)
y=x^2 +4

Answer 1

1.
`y = \sqrt[3]{x-5}`
The inverse function can be found by interchanging x and y, then solving for y.

`x = \sqrt[3]{y-5}`

`x^(3) = y - 5`

`y = x^(3) + 5`
This IS a function.


2. y = x^2 +4
.. x = y^2 +4
.. x -4 = y^2
.. y = ±√(x -4) . . . . . . the inverse relation is double-valued, so is NOT a function

_____
In general, odd degree polynomials and roots have inverses; even degree polynomials do not. Even-degree roots will typically be double-valued, so an inverse function can be defined for one or the other of the values, but not both.

In the above case,
.. y = √(x -4) is a function, applicable only for y ≥ 0.