Answer:
[tex]f^{-1}(x) = \sqrt[3]{\frac{x -4}{5}} +2[/tex]
Step-by-step explanation:
Given
[tex]f(x) = 5(x - 2)^3 + 4[/tex]
Required
Determine the inverse
[tex]f(x) = 5(x - 2)^3 + 4[/tex]
Replace f(x) with y
[tex]y = 5(x - 2)^3 + 4[/tex]
Swap the positions of x and y
[tex]x = 5(y - 2)^3 + 4[/tex]
Make y the subject
[tex]x -4= 5(y - 2)^3[/tex]
Divide by 5
[tex]\frac{x -4}{5}= (y - 2)^3[/tex]
Take cube roots of both sides
[tex]\sqrt[3]{\frac{x -4}{5}}= y - 2[/tex]
Add 2 to both sides
[tex]\sqrt[3]{\frac{x -4}{5}} +2 = y[/tex]
[tex]y = \sqrt[3]{\frac{x -4}{5}} +2[/tex]
Replace y with [tex]f^{-1}(x)[/tex]
[tex]f^{-1}(x) = \sqrt[3]{\frac{x -4}{5}} +2[/tex]
