To find the inverse [tex]\( f^{-1}(x) \)[/tex] of the function [tex]\( f(x) = x^3 + 2 \)[/tex], we need to follow a series of steps. Let’s work through this problem systematically.
1. Write the function with [tex]\( y \)[/tex] replacing [tex]\( f(x) \)[/tex]:
[tex]\[
y = x^3 + 2
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
y = x^3 + 2
\][/tex]
Subtract 2 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
y - 2 = x^3
\][/tex]
3. Take the cube root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \sqrt[3]{y - 2}
\][/tex]
4. Express the function [tex]\( f^{-1}(x) \)[/tex]:
To find the inverse, we swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This gives us the inverse function in terms of [tex]\( x \)[/tex]:
[tex]\[
f^{-1}(x) = \sqrt[3]{x - 2}
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = (x - 2)^{1/3}
\][/tex]
