Answer:
[tex] r^{-1}(x) = \sqrt[3]{(x - 2)^3 + 5} [/tex]
Step-by-step explanation:
To find the inverse function [tex] r^{-1}(x) [/tex] for the given function [tex] r(x) = (x^3 - 5)^{\frac{1}{3}} + 2 [/tex], we follow these steps:
Replace [tex] r(x) [/tex] with [tex] y [/tex]:
[tex] y = (x^3 - 5)^{\frac{1}{3}} + 2 [/tex]
Swap [tex] x [/tex] and [tex] y [/tex] to express the inverse function:
[tex] x = (y^3 - 5)^{\frac{1}{3}} + 2 [/tex]
Solve for [tex] y [/tex]:
[tex] x - 2 = (y^3 - 5)^{\frac{1}{3}} [/tex]
Cube both sides to eliminate the cube root:
[tex] (x - 2)^3 = y^3 - 5 [/tex]
Add 5 to both sides:
[tex] (x - 2)^3 + 5 = y^3 [/tex]
Take the cube root of both sides:
[tex] \sqrt[3]{(x - 2)^3 + 5} = y [/tex]
Therefore, the inverse function [tex] r^{-1}(x) [/tex] is given by:
[tex] r^{-1}(x) = \sqrt[3]{(x - 2)^3 + 5} [/tex]
So, the function has an inverse function, and its formula is:
[tex] r^{-1}(x) = \sqrt[3]{(x - 2)^3 + 5} [/tex]
