The problem asks to find $g(-1)$ for a function $f(x)$ after it has been rotated 180° around the origin. The function given is:

$f(x) = x^3 - x^2 - 2x$

Step-by-Step Solution:

1. Rotation by 180° Transformation: When a function $f(x)$ is rotated by 180° around the origin, the new function $g(x)$ is given by:

   $g(x) = -f(-x)$

2. Finding $g(x)$: We need to compute $-f(-x)$.

   First, substitute $-x$ into $f(x)$: $f(-x) = (-x)^3 - (-x)^2 - 2(-x) = -x^3 - x^2 + 2x$

   Now, multiply by $-1$: $g(x) = -f(-x) = x^3 + x^2 - 2x$

3. Finding $g(-1)$: Substitute $-1$ into $g(x)$: $g(-1) = (-1)^3 + (-1)^2 - 2(-1)$ Simplifying step-by-step: $g(-1) = -1 + 1 + 2 = 2$

So, the value of $g(-1)$ is:

$g(-1) = 2$

Would you like more details or have any questions?

Related Questions:

1. What is the transformation formula for a 90° rotation of a function around the origin?
2. How would $f(x)$ change if reflected over the x-axis?
3. Can you find $g(1)$ for the rotated function?
4. What is the derivative of $f(x)$ and how does it change under rotation?
5. How would the function change if rotated around a different point, such as $(1, 1)$?

Tip: When rotating a function, remember to adjust both the x and y components according to the rotation rules to find the new function accurately.