To find the inverse of the function [tex]\( f(x) = \sqrt[3]{8x} + 4 \)[/tex], follow these steps. Remember that the inverse function [tex]\( f^{-1}(x) \)[/tex] will switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the equation.
Given:
[tex]\[ f(x) = \sqrt[3]{8x} + 4 \][/tex]
Step 1: Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex] for simplicity:
[tex]\[ y = \sqrt[3]{8x} + 4 \][/tex]
Step 2: To find the inverse, we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. First, isolate the cube root term by subtracting 4 from both sides:
[tex]\[ y - 4 = \sqrt[3]{8x} \][/tex]
Step 3: To eliminate the cube root, cube both sides of the equation:
[tex]\[ (y - 4)^3 = 8x \][/tex]
Step 4: Solve for [tex]\( x \)[/tex] by dividing both sides by 8:
[tex]\[ x = \frac{(y - 4)^3}{8} \][/tex]
Step 5: Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function:
[tex]\[ f^{-1}(x) = \frac{(x - 4)^3}{8} \][/tex]
Thus, the inverse function of [tex]\( f(x) = \sqrt[3]{8x} + 4 \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{(x - 4)^3}{8} \][/tex]
This completes the derivation of the inverse function.
