To find the inverse of the function [tex]\( y = 3x^3 - 5 \)[/tex], we need to switch the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and then solve for [tex]\( y \)[/tex]. Here are the steps:
1. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] and [tex]\( x \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
x = 3y^3 - 5
\][/tex]
2. Solve for [tex]\( y \)[/tex]:
First, isolate the term involving [tex]\( y \)[/tex]:
[tex]\[
x + 5 = 3y^3
\][/tex]
Next, divide both sides by 3 to completely isolate [tex]\( y^3 \)[/tex]:
[tex]\[
\frac{x + 5}{3} = y^3
\][/tex]
Now, take the cube root of both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \sqrt[3]{\frac{x + 5}{3}}
\][/tex]
So the inverse function is [tex]\( y = \sqrt[3]{\frac{x + 5}{3}} \)[/tex].
Now, let's verify which option matches this inverse function:
(A) [tex]\( y = \sqrt[3]{\frac{x + 5}{3}} \)[/tex]
(B) [tex]\( y = \sqrt[3]{\frac{x-5}{3}} \)[/tex]
(C) [tex]\( y = 3 \sqrt[3]{x-5} \)[/tex]
(D) [tex]\( y = \sqrt[3]{\frac{x}{3}} + 5 \)[/tex]
By comparing these options to our derived inverse function [tex]\( y = \sqrt[3]{\frac{x + 5}{3}} \)[/tex], it’s clear that option (A) is correct.
Therefore, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
