Final answer:
The rule for the function g(x), defined as -f(x-1), where f(x) = x^3 - 4x^2 + 6, is g(x) = -x^3 + 7x^2 - 11x - 1 after applying the transformation and simplifying the expression.
Step-by-step explanation:
To write a rule for g(x), which is defined as -f(x-1), we first need to apply the transformation to the function f(x). The function f(x) is given as
f(x)=x^3 - 4x^2 + 6.
We replace every instance of x in f(x) with (x - 1) to find f(x - 1), that is f(x-1) = (x-1)^3 - 4(x-1)^2 + 6. We then simplify this expression:
f(x - 1) = (x - 1)^3 - 4(x - 1)^2 + 6
= (x^3 - 3x^2 + 3x - 1) - 4(x^2 - 2x + 1) + 6
= x^3 - 3x^2 + 3x - 1 - 4x^2 + 8x - 4 + 6
= x^3 - 7x^2 + 11x + 1
Now, to find g(x), we multiply this expression by -1:
g(x) = -f(x - 1)
= -(x^3 - 7x^2 + 11x + 1)
= -x^3 + 7x^2 - 11x - 1
Therefore, the rule for g(x) is g(x) = -x^3 + 7x^2 - 11x - 1.