Final answer:
To find the rule for g(x), you substitute (x-1) into f(x) = x³ - 4x² + 6, simplify the expression, make it negative, and then add 6. The resulting expression is the rule for the function g(x).
Step-by-step explanation:
The goal of this problem is to find a rule for the function g(x) based on the given information about the functions f(x) and g(x). We are given f(x) = x³ - 4x² + 6. To create g(x), we first need to adjust the function f to f(x-1), which involves substituting (x-1) for every x in the original function f. Once we have the result of f(x-1), we have to take the negative of this result and then add 6 to get g(x).
Following these steps:
- Find f(x-1) which means plugging in (x-1) into f: f(x-1) = (x-1)³ - 4(x-1)² + 6.
- Expand f(x-1) and simplify.
- Since g(x) = -f(x-1) + 6, take the negative of the result from step 2 and add 6.
Once all the terms have been combined and simplified, this will give us the final rule for g(x).