To findTo the find inverse of the relation $y the = (3x + 2)^{1/3} + 2x$, inverse we need to express of the x relation \ \() y in = terms ( of3 \(x + y2).

^{Given1: /[ 3}y + = (23xx \ +), follow2 these) steps^{1:

/3} + 21. **xSwap ( x ) and\ ( y]

To find \ the): inverse**:

1 . ( Swap x ( = ( x \3)y and + ( y2)^{ \1/3}):

• \2[ yx \ =)

(32y +. 2) **^{1Is/3}olate + 2 they \ cube]

2. root Now, term:** we need
   to solve for ( y \ (). Rearranging x terms: -$$x - 22y =y ( =3 (y3 +y + 2)2^{)^{11//3} \3)

3.} **\Cube]

both sides:**
3 ( (.x - 2y Cube)^3 both = sides3 toy eliminate the + cube root: \2[ (x - 2)

y4)^3 = .3y + ** Re2arr$$

ange4. Expand to the left-hand solve side:

8y^3 = 3y \( + 2$$ 5.3 Rearryanging all terms = involving $$y ($$ onx one side: \ -[ 82yy^)^33 - - 2 \)12 \(xy^ y2 = + \ (6xfrac^{(x2 - - 2y)^33 - )y2 -}{ x3^}3 \ +) This2 equation = is implicit0 and \ complex,] This is a making cubic equation it in \( challenging y to isolate \ \( y) \ and) explicitly. does In not this case have, the relation a may not simple have a closed simple-form explicit solution inverse.. Generally, solvingWould cubic you equations like requires a numerical numerical methods approach or to specific find formulas values ( ofCard theano inverse's or method further), explanation which are complex on. theTherefore steps,? the inverse relation Let is given implicitly me know by: if\ you[ have8 anyy^ questions3 - ! 12###xy^2 Related + Questions (: 1.6 What are thex general conditions for^ a2 function to have - an inverse3? )y2 -. x How^ do3 you + verify if2 a = function is one -to-one (0bij ective\)? ] 3Would you. like Can further you details find or the have derivative any of questions the? original functionHere are to some analyze related its monot questions for explorationonicity: ? 41. What. is the How graphical interpretation of do finding we an inverse determine if function a? function5 has. an How inverse do? 2. transformations What affect are the the general inverse of methods for a finding function invers? es of** cubicTip equations:**? Always3. How can check we use if graphical methods a to find function inverses of functions is? 4. one What-to are-one the before properties attempting of inverse to functions find? 5 its. inverse How; does this the ensures domain and that range the of inverse a exists function. change when finding its inverse? **Tip:** To check if a function has an inverse, ensure it passes the Horizontal Line Test—meaning every horizontal line intersects the graph at most once!