Find the inverse of $f(x)=-7 x^{3}+12$

Step 1 ：First, we need to understand the problem. We are asked to find the inverse of the function \(f(x) = -7x^3 + 12\). The inverse of a function 'undoes' the operation of the original function.

Step 2 ：To find the inverse of a function, we first replace the function notation \(f(x)\) with \(y\). So, we have \(y = -7x^3 + 12\).

Step 3 ：Next, we swap \(x\) and \(y\). This gives us \(x = -7y^3 + 12\).

Step 4 ：Now, we solve this equation for \(y\). First, we subtract 12 from both sides to get \(x - 12 = -7y^3\).

Step 5 ：Then, we divide both sides by -7 to get \(\frac{x - 12}{-7} = y^3\).

Step 6 ：Finally, we take the cube root of both sides to solve for \(y\). This gives us \(y = \sqrt[3]{\frac{x - 12}{-7}}\).

Step 7 ：So, the inverse of the function \(f(x) = -7x^3 + 12\) is \(f^{-1}(x) = \sqrt[3]{\frac{x - 12}{-7}}\).

Step 8 ：We can check our answer by substituting \(f^{-1}(x)\) into the original function and simplifying. If we get \(x\), then our answer is correct.

Step 9 ：Substituting \(f^{-1}(x)\) into the original function, we get \(f(f^{-1}(x)) = -7(\sqrt[3]{\frac{x - 12}{-7}})^3 + 12\).

Step 10 ：Simplifying this expression, we get \(f(f^{-1}(x)) = -7(\frac{x - 12}{-7}) + 12 = x\).

Step 11 ：Since \(f(f^{-1}(x)) = x\), our answer is correct. The inverse of the function \(f(x) = -7x^3 + 12\) is \(\boxed{f^{-1}(x) = \sqrt[3]{\frac{x - 12}{-7}}}\).