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.

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Work Shown:

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

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

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Side note:

If $f(x) = \frac{1}{4}x^3 - 2$ 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.