Given:
The function is [tex]f(x)=3(x+7)^{\frac{1}{4}}[/tex].
To find:
The function [tex]f^{-1}(x)[/tex].
Solution:
We have,
[tex]f(x)=3(x+7)^{\frac{1}{4}}[/tex]
Substitute f(x)=y.
[tex]y=3(x+7)^{\frac{1}{4}}[/tex]
Interchange x and y.
[tex]x=3(y+7)^{\frac{1}{4}}[/tex]
Divide both sides by 3.
[tex]\dfrac{x}{3}=(y+7)^{\frac{1}{4}}[/tex]
Taking power 4 on both sides.
[tex]\left(\dfrac{x}{3}\right)^4=y+7[/tex]
Subtract 7 from both sides.
[tex]\left(\dfrac{x}{3}\right)^4-7=y[/tex]
[tex]y=\left(\dfrac{x}{3}\right)^4-7[/tex]
Substitute [tex]y=f^{-1}(x)[/tex].
[tex]f^{-1}(x)=\left(\dfrac{x}{3}\right)^4-7[/tex]
Therefore, the correct option is C.
