Question

Determine the equation of the inverse of y = 1/4 x^3 - 2

Answer 1

Answer:
`y = \sqrt[3]{4x+8}`

All of 4x+8 is under a cube root sign.

=====================================================

Work Shown:

To find the inverse, we swap x and y, then solve for y.

`y = (1)/(4)x^3 - 2\\\\x = (1)/(4)y^3 - 2\\\\x+2 = (1)/(4)y^3\\\\4(x+2) = y^3\\\\4x+8 = y^3\\\\y^3 = 4x+8\\\\y = \sqrt[3]{4x+8}\\\\`

------------

Side note:

If
`f(x) = (1)/(4)x^3 - 2` and
`g(x) = \sqrt[3]{4x+8}`, then
`f(g(x)) = x` and
`g(f(x)) = x`for all x values in the domain. Effectively, you use function composition to confirm that we have the correct inverse equation.