Final answer:
To find f^-1(x), we need to solve the equation g(f(x)) = x.
We substitute the expression for f(x) into g(f(x)), simplify, and solve for f^-1(x).
Step-by-step explanation:
To find f-1(x), we need to solve the equation g(f(x)) = x.
Let's substitute the expression for f(x) into g(f(x)).
We have
- g(f(x)) = (x - 2x2) - 2(x - 2x2)2.
Simplifying, we get
- g(f(x)) = x - 2x2 - 2(x2 - 4x3 + 4x4).
Further simplifying, we have
- g(f(x)) = x - 2x2 - 2x2 + 8x3 - 8x4.
Combining like terms, we get
- g(f(x)) = -3x2 + 8x3 - 8x4 + x.
Finally, we set this expression equal to x and solve for f-1(x). This involves finding the inverse of a polynomial function.
Learn more about Finding the inverse of a function