To find the value of [tex]\((g \circ f)(4)\)[/tex], we need to follow a few steps to determine the result.
### Step 1: Calculate [tex]\(f(4)\)[/tex]
Given the function [tex]\(f(x) = -x^3\)[/tex], we need to substitute [tex]\(x = 4\)[/tex] into this function:
[tex]\[
f(4) = -(4)^3 = -64
\][/tex]
So, [tex]\(f(4) = -64\)[/tex].
### Step 2: Calculate [tex]\(g(f(4))\)[/tex]
Next, we use the value obtained from [tex]\(f(4)\)[/tex] as the input for the function [tex]\(g\)[/tex]. The function [tex]\(g(x)\)[/tex] is defined as:
[tex]\[
g(x) = \left| \frac{1}{8}x - 1 \right|
\][/tex]
Here, we substitute [tex]\(x = -64\)[/tex] into the function [tex]\(g\)[/tex]:
[tex]\[
g(-64) = \left| \frac{1}{8} \cdot (-64) - 1 \right|
\][/tex]
Perform the multiplication first:
[tex]\[
\frac{1}{8} \cdot (-64) = -8
\][/tex]
Then, substitute back into the equation:
[tex]\[
g(-64) = \left| -8 - 1 \right|
\][/tex]
Simplify inside the absolute value:
[tex]\[
g(-64) = \left| -9 \right| = 9
\][/tex]
So, [tex]\(g(f(4)) = 9\)[/tex].
### Conclusion
The value of [tex]\((g \circ f)(4)\)[/tex] is [tex]\(\boxed{9}\)[/tex].
