Question

What is the inverse of the function f(x) = -1/2√x³, x≥ -3?

Answer 1

To find the inverse of the function f(x) = -1/2√x³, swap x and y, then solve for y by performing a series of algebraic manipulations. The inverse function is f-1(x) = -8x.

Step-by-step explanation:

To find the inverse of the function f(x) = -1/2√x³, where x≥ -3, we need to switch the roles of x and y, and then solve for y. The process is as follows:

1. 
   
3. Start with the original function, replacing f(x) with y: y = -1/2√x³.
4. 
   
6. Swap x and y: x = -1/2√y³.
7. 
   
9. Multiply both sides by -2 to isolate the cubic root: -2x = √y³.
10. 
    
12. Remove the cubic root by raising both sides to the power of 3: (-2x)³ = y³.
13. 
    
15. Now solve for y by cube rooting both sides: y = ∓((-2x)³).
16. 
    
18. Simplify the cube: y = (-2)³ * ∓(x³) which simplifies to y = -8x.

The inverse function is therefore f-1(x) = -8x.