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Consider function [tex]$f$[/tex].
[tex]\[ f(x) = \sqrt[3]{8x} + 4 \][/tex]

To determine the inverse of function [tex]$f$[/tex], change [tex]$f(x)$[/tex] to [tex][tex]$y$[/tex][/tex], switch [tex]$x$[/tex] and [tex]$y$[/tex], and solve for [tex][tex]$y$[/tex][/tex].

The resulting function can be written as:
[tex]\[ f^{-1}(x) = (x - \quad)^3 \][/tex]



Answer :

To determine the inverse of the function [tex]\( f(x) = \sqrt[3]{8x} + 4 \)[/tex], let's proceed step by step.

1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\( y = \sqrt[3]{8x} + 4 \)[/tex]

2. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\( x = \sqrt[3]{8y} + 4 \)[/tex]

3. Solve for [tex]\( y \)[/tex]:
- First, subtract 4 from both sides to isolate the cube root term:
[tex]\( x - 4 = \sqrt[3]{8y} \)[/tex]

- Next, cube both sides to eliminate the cube root:
[tex]\( (x - 4)^3 = 8y \)[/tex]

- Finally, divide both sides by 8 to solve for [tex]\( y \)[/tex]:
[tex]\( y = \frac{(x - 4)^3}{8} \)[/tex]

4. Rewriting the function:
The inverse function can be written as:
[tex]\( f^{-1}(x) = \frac{(x - 4)^3}{8} \)[/tex]

Thus, the correct placement for the missing number is:

The resulting function can be written as
[tex]\[ f^{-1}(x) = \frac{(x - 4)^3}{8} \][/tex]