To find the inverse of the function [tex]\( f(x) = \sqrt[3]{x - 2} \)[/tex], we need to follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt[3]{x - 2}
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
x = \sqrt[3]{y - 2}
\][/tex]
3. Isolate [tex]\( y \)[/tex] by undoing the cube root. To do this, cube both sides of the equation:
[tex]\[
x^3 = y - 2
\][/tex]
4. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = x^3 + 2
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = x^3 + 2
\][/tex]
So the inverse of the function [tex]\( f(x) = \sqrt[3]{x - 2} \)[/tex] is:
[tex]\[
f^{-1}(x) = x^3 + 2
\][/tex]
