To find the inverse of the function [tex]\( f(x) = -7x^3 + 12 \)[/tex], let's denote the inverse function by [tex]\( f^{-1}(y) \)[/tex]. This means that if [tex]\( y = f(x) \)[/tex], then [tex]\( x = f^{-1}(y) \)[/tex]. Our goal is to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] through the following steps:
1. Start with the original function and set it equal to [tex]\( y \)[/tex]:
[tex]\[
y = -7x^3 + 12
\][/tex]
2. Subtract 12 from both sides to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
y - 12 = -7x^3
\][/tex]
3. Divide both sides by -7 to further isolate [tex]\( x^3 \)[/tex]:
[tex]\[
\frac{y - 12}{-7} = x^3
\][/tex]
4. Simplify the left-hand side:
[tex]\[
x^3 = \frac{12 - y}{7}
\][/tex]
5. Take the cube root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \sqrt[3]{\frac{12 - y}{7}}
\][/tex]
Thus, the inverse function [tex]\( f^{-1}(y) \)[/tex] is:
[tex]\[
f^{-1}(y) = \sqrt[3]{\frac{12 - y}{7}}
\][/tex]
Given the multiple-choice options provided, we can match this derived inverse function:
- [tex]\( f^{-1}(x) = \sqrt[3]{\frac{-x + 12}{7}} \)[/tex]
- [tex]\( f^{-1}(x) = \frac{1}{-7 x^3 + 12} \)[/tex]
- [tex]\( f^{-1}(x) = 7 x^3 - 12 \)[/tex]
- [tex]\( f^{-1}(x) = \sqrt[3]{\frac{x - 12}{7}} \)[/tex]
The correct choice is:
[tex]\[
f^{-1}(x) = \sqrt[3]{\frac{12 - x}{7}}
\][/tex]
This matches with the first option if we understand that [tex]\( y \)[/tex] in our derived expression should be replaced with [tex]\( x \)[/tex] to fit the inverse function notation [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = \sqrt[3]{\frac{12 - x}{7}}
\][/tex]
Therefore, the correct answer is:
[tex]\[ f^{-1}(x) = \sqrt[3]{\frac{-x + 12}{7}} \][/tex]
