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]
