To find the inverse of the function [tex]\( f(x) = \sqrt[3]{x-2} \)[/tex], follow these steps:
1. Rewrite the function:
Instead of [tex]\( f(x) \)[/tex], write the function using [tex]\( y \)[/tex] to make manipulation easier:
[tex]\[
y = \sqrt[3]{x - 2}
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
Isolate [tex]\( x \)[/tex] on one side of the equation. To do this, first eliminate the cube root by cubing both sides of the equation:
[tex]\[
y^3 = x - 2
\][/tex]
3. Isolate [tex]\( x \)[/tex]:
Add 2 to both sides of the equation to solve for [tex]\( x \)[/tex]:
[tex]\[
y^3 + 2 = x
\][/tex]
4. Write the inverse function:
Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function. The inverse of [tex]\( f \)[/tex] is often denoted as [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = x^3 + 2
\][/tex]
Thus, the inverse of the function [tex]\( f(x) = \sqrt[3]{x-2} \)[/tex] is:
[tex]\[
\boxed{x^3 + 2}
\][/tex]
