What is the inverse of the function y = (x - 2)^(3/2)?

To find the inverse of the function ( y = (x - 2)^{3/2} ), we first need to express ( x ) in terms of ( y ) by swapping the roles of ( x ) and ( y ) and then solving for ( y ).

Starting from the original equation:

[ y = (x - 2)^{3/2} ]

We swap ( y ) and ( x ):

[ x = (y - 2)^{3/2} ]

Next, we raise both sides to the power of ( \frac{2}{3} ) to eliminate the exponent on the right:

[ x^{2/3} = y - 2 ]

Now, we solve for ( y ):

[ y = x^{2/3} + 2 ]

To understand why the choice ( y = \text{cbrt}(x^2) + 2 ) is the correct answer, we can rewrite it. The cube root function, commonly denoted as ( \text{cbrt}(x) ) or ( x^{1/3} ), relates to the exponent we derived during the inverse calculation