To determine the inverse of the function [tex]\( f(x) = x^3 - 5 \)[/tex], we must find a function [tex]\( f^{-1}(x) \)[/tex] such that when [tex]\( f(f^{-1}(x)) \)[/tex] and [tex]\( f^{-1}(f(x)) \)[/tex] are applied, we get [tex]\( x \)[/tex] back.
Let's start by setting [tex]\( y = f(x) \)[/tex]:
[tex]\[ y = x^3 - 5 \][/tex]
To find the inverse function, we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = x^3 - 5 \][/tex]
First, add 5 to both sides to isolate the cubic term:
[tex]\[ y + 5 = x^3 \][/tex]
Next, take the cube root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt[3]{y + 5} \][/tex]
Thus, the inverse function is:
[tex]\[ f^{-1}(x) = \sqrt[3]{x + 5} \][/tex]
So, the correct inverse function of [tex]\( f(x) = x^3 - 5 \)[/tex] is:
[tex]\[ f^{-1}(x) = \sqrt[3]{x + 5} \][/tex]
Among the given options, this matches:
[tex]\[ f^{-1}(x) = \sqrt[3]{x + 5} \][/tex]
