Question

which function is the inverse of :
$g(x) = \sqrt3{2x - 6} + 4$ ?
$\bigcirc$ $f(x) = \frac{(x - 4)^3}{2} + 3$
$\bigcirc$ $f(x) = \frac{(x - 2)^3}{64} + 3$
$\bigcirc$ $f(x) = \frac{(x - 2)^3}{4} + 3$
$\bigcirc$ $f(x) = \frac{(x - 4)}{8} + 3$

Explanation:

Step1: Replace $g(x)$ with $y$

$y = \sqrt[3]{2x - 6} + 4$

Step2: Swap $x$ and $y$

$x = \sqrt[3]{2y - 6} + 4$

Step3: Isolate the cube root

$x - 4 = \sqrt[3]{2y - 6}$

Step4: Cube both sides

$(x - 4)^3 = 2y - 6$

Step5: Solve for $y$

$2y = (x - 4)^3 + 6$
$y = \frac{(x - 4)^3}{2} + 3$

Answer:

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