To find the inverse of the function [tex]\( f(x) = 3x^3 - 4 \)[/tex], we need to follow a systematic approach:
1. Start by replacing [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = 3x^3 - 4
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
- Step 1: Add 4 to both sides of the equation:
[tex]\[
y + 4 = 3x^3
\][/tex]
- Step 2: Divide both sides by 3:
[tex]\[
\frac{y + 4}{3} = x^3
\][/tex]
- Step 3: Take the cube root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \left( \frac{y + 4}{3} \right)^{\frac{1}{3}}
\][/tex]
3. Replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to express the inverse function [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = \left( \frac{x + 4}{3} \right)^{\frac{1}{3}}
\][/tex]
Thus, the inverse function is [tex]\( f^{-1}(x) = \left( \frac{x + 4}{3} \right)^{\frac{1}{3}} \)[/tex].
Comparing this with the given options:
- A) [tex]\( f^{-1}(x) = \frac{x+1}{3} \)[/tex]
- B) [tex]\( f^{-1}(x) = \sqrt[2]{\frac{x+4}{3}} \)[/tex]
- C) [tex]\( f^{-1}(x) = 3x + 4 \)[/tex]
- D) [tex]\( f^{-1}(x) = \sqrt{\frac{x+4}{3}} \)[/tex]
We see that none of the options exactly match the derived inverse. The correct expression derived for the inverse function does not perfectly align with any of the provided multiple choices.
