Let's evaluate the function [tex]\( g(x) = \left| x^3 - 3x^2 + 3x - 1 \right| \)[/tex] at [tex]\( x = -4 \)[/tex].
First, we will substitute [tex]\( x = -4 \)[/tex] into the polynomial inside the absolute value:
[tex]\[
g(-4) = \left| (-4)^3 - 3(-4)^2 + 3(-4) - 1 \right|
\][/tex]
Next, we will evaluate each term individually:
[tex]\[
(-4)^3 = -64
\][/tex]
[tex]\[
3(-4)^2 = 3 \times 16 = 48
\][/tex]
[tex]\[
3(-4) = -12
\][/tex]
[tex]\[
-1 \text{ is a constant term}
\][/tex]
Now, putting all these together:
[tex]\[
g(-4) = \left| -64 - 48 - 12 - 1 \right|
\][/tex]
Combine the terms inside the absolute value:
[tex]\[
g(-4) = \left| -64 - 48 - 12 - 1 \right|
\][/tex]
[tex]\[
g(-4) = \left| -125 \right|
\][/tex]
The absolute value of [tex]\(-125\)[/tex] is:
[tex]\[
\left| -125 \right| = 125
\][/tex]
Therefore, the value of the function [tex]\( g(x) \)[/tex] at [tex]\( x = -4 \)[/tex] is:
[tex]\[
g(-4) = 125
\][/tex]
