Question

question 1
which function is the inverse of:
$g(x) = \frac{(x - 3)^3}{2} - 2$ ?
$\circ$ $f(x) = 2\sqrt3{x + 3} + 1$
$\circ$ $f(x) = \sqrt3{2x + 4} + 3$
$\circ$ $f(x) = \sqrt3{2x + 6} - 2$
$\circ$ $f(x) = 2\sqrt3{x + 1} + 3$

Explanation:

Step1: Set $y = g(x)$

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

Step2: Swap $x$ and $y$

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

Step3: Isolate the cubic term

Add 2 to both sides:
$x + 2 = \frac{(y-3)^3}{2}$
Multiply by 2:
$2(x + 2) = (y-3)^3$
Simplify:
$2x + 4 = (y-3)^3$

Step4: Solve for $y$

Take cube root of both sides:
$\sqrt[3]{2x + 4} = y - 3$
Add 3 to both sides:
$y = \sqrt[3]{2x + 4} + 3$

Answer:

$\boldsymbol{f(x) = \sqrt[3]{2x + 4} + 3}$ (the second option)