Answer :
To find the inverse function of [tex]\( f(x) = x^3 \)[/tex], we follow these steps:
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = x^3 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse:
[tex]\[ x = y^3 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = x^{1/3} \][/tex]
4. Since we denote the inverse function as [tex]\( f^{-1}(x) \)[/tex], we have:
[tex]\[ f^{-1}(x) = -1 \][/tex]
So, the inverse function [tex]\( f^{-1}(x) \)[/tex] is [tex]\(\boxed{-1}\)[/tex].
1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = x^3 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse:
[tex]\[ x = y^3 \][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = x^{1/3} \][/tex]
4. Since we denote the inverse function as [tex]\( f^{-1}(x) \)[/tex], we have:
[tex]\[ f^{-1}(x) = -1 \][/tex]
So, the inverse function [tex]\( f^{-1}(x) \)[/tex] is [tex]\(\boxed{-1}\)[/tex].
