To find the inverse function of [tex]\( f(x) = 3 + \sqrt[3]{x} \)[/tex], follow these steps:
1. Start by expressing [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = 3 + \sqrt[3]{x}
\][/tex]
2. Isolate [tex]\( \sqrt[3]{x} \)[/tex] on one side:
[tex]\[
y - 3 = \sqrt[3]{x}
\][/tex]
3. Cube both sides to eliminate the cube root:
[tex]\[
(y - 3)^3 = x
\][/tex]
4. Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
x = (y - 3)^3
\][/tex]
5. Interpret this expression as the inverse function [tex]\( f^{-1}(y) \)[/tex]:
[tex]\[
f^{-1}(y) = (y - 3)^3
\][/tex]
6. Substitute [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to match the standard notation for functions:
[tex]\[
f^{-1}(x) = (x - 3)^3
\][/tex]
Therefore, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = (0.333333333333333x - 1)^3
\][/tex]
