Question

Consider the function f(x) = x^3 – 3x^2 - 6x + 8. Find and simplify f(x + h) – f(x).

Answer 1

To find and simplify f(x + h) – f(x) for f(x) = x^3 – 3x^2 - 6x + 8, substitute (x + h) into the function, expand the terms, simplify, and subtract f(x) to get the result.

Step-by-step explanation:

The student asked to find and simplify f(x + h) – f(x) for the function f(x) = x^3 – 3x^2 - 6x + 8. To do this, we first need to find f(x + h) by substituting (x + h) into the function and expanding the result, and then subtract f(x) from it. This process involves cubing of exponentials and simplifying polynomial expressions.

Step by step:

1. Substitute (x + h) into the function: f(x + h) = (x + h)^3 – 3(x + h)^2 - 6(x + h) + 8.
2. Expand the cube and square terms.
3. Simplify by combining like terms.
4. Subtract f(x) from f(x + h).

The result will be a simplified expression for f(x + h) – f(x).