To find the inverse of the function [tex]\( f(x) = \sqrt[3]{x+10} \)[/tex], we need to determine a function [tex]\( f^{-1}(x) \)[/tex] such that [tex]\( f(f^{-1}(x)) = x \)[/tex]. Here is a step-by-step process to find the inverse:
1. Start with the function definition:
[tex]\[
f(x) = \sqrt[3]{x + 10}
\][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt[3]{x + 10}
\][/tex]
3. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]: This is done because we are looking for the inverse, which involves solving for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
x = \sqrt[3]{y + 10}
\][/tex]
4. Solve the equation for [tex]\( y \)[/tex]: We need to isolate [tex]\( y \)[/tex] on one side of the equation.
[tex]\[
x = (y + 10)^{1/3}
\][/tex]
To remove the cube root, cube both sides of the equation:
[tex]\[
x^3 = (y + 10)
\][/tex]
5. Isolate [tex]\( y \)[/tex]:
[tex]\[
y = x^3 - 10
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[
f^{-1}(x) = x^3 - 10
\][/tex]
This inverse function will map any output [tex]\( y \)[/tex] of the original function [tex]\( f \)[/tex] back to the original input [tex]\( x \)[/tex]. Therefore, the inverse of [tex]\( f(x) = \sqrt[3]{x+10} \)[/tex] is:
[tex]\[
f^{-1}(x) = x^3 - 10
\][/tex]
