Given f(x)=6x^4, find f^-1(x). Then state whether f^-1(x) is a function.
Accepted Solution
A:
Answer:[tex]f^{-1}(x)=(\frac{x}{6})^{\frac{1}{4}}[/tex]This is a function for [0,∞).Step-by-step explanation:The given function is[tex]f(x)=6x^4[/tex]We need to find the [tex]f^{-1}(x)[/tex].Step 1: Replace f(x) be y.[tex]y=6x^4[/tex]Step 2: Interchange x and y.[tex]x=6y^4[/tex]Step 3: Isolate variable y.[tex]\frac{x}{6}=y^4[/tex][tex](\frac{x}{6})^{\frac{1}{4}}=y[/tex]Step 4: Interchange the sides.[tex]y=(\frac{x}{6})^{\frac{1}{4}}[/tex]Step 5: Replace y by [tex]f^{-1}(x)[/tex].[tex]f^{-1}(x)=(\frac{x}{6})^{\frac{1}{4}}[/tex]Therefore, [tex]f^{-1}(x)=(\frac{x}{6})^{\frac{1}{4}}[/tex].This function is is defined for all positive values of x.The inverse of function [tex]f(x)=6x^4[/tex] is a function for [0,∞).