To determine the inverse of the function [tex]\( f(x) = \sqrt[2]{8x} + 4 \)[/tex], we need to go through the following detailed steps:
1. Rewrite [tex]\( f(x) \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt[2]{8x} + 4
\][/tex]
2. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse:
[tex]\[
x = \sqrt[2]{8y} + 4
\][/tex]
3. Isolate the square root term:
[tex]\[
x - 4 = \sqrt[2]{8y}
\][/tex]
4. Square both sides to get rid of the square root:
[tex]\[
(x - 4)^2 = 8y
\][/tex]
5. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{(x - 4)^2}{8}
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] in terms of [tex]\( y \)[/tex] is:
[tex]\[
f^{-1}(x) = \frac{(x - 4)^2}{8}
\][/tex]
To match the form given in the problem [tex]\( f^{-1}(x) = (x - \quad)^3 \)[/tex], we need to identify what [tex]\( (x - 4)^2 / 8 \)[/tex] would be equivalent to if written in another form.
Given the structure of the problem, the blank [tex]\((x - \quad)^3\)[/tex] should correspond to shifting the value used in the transformation. The inverse function after solving shows the shift was by the value 4.
Therefore:
[tex]\[
f^{-1}(x)=(x-4)^3
\][/tex]
The missing value in the blank is [tex]\( 4 \)[/tex].
