To determine the inverse of the function [tex]\( f(x) = \sqrt[3]{8x} + 4 \)[/tex], follow these steps:
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, 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, solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{(x - 4)^3}{8} \][/tex]
So, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{(x - 4)^3}{8} \][/tex]
In simplified form, this can also be written as:
[tex]\[ f^{-1}(x) = 8 \left( \frac{x - 4}{8} \right)^3 = 8 \left(0.25 (x - 4)\right)^3 \][/tex]
Given the previously determined result:
[tex]\[ f^{-1}(x) = 8 \left(0.25 (x - 4)\right)^3 \][/tex]
Thus, we can see that:
[tex]\[ f^{-1}(x) = (x - 4)^3 \][/tex]
So the final answer is:
The resulting function can be written as:
[tex]\[ f^{-1}(x) = (x - 4)^3. \][/tex]
