1. **State the problem:** Find the inverse of the function $$y = \sqrt[4]{x} - 2$$. 2. **Recall the definition of inverse functions:** The inverse function swaps the roles of $$x$$ and $$y$$. To find the inverse, we solve for $$x$$ in terms of $$y$$ and then interchange the variables. 3. **Start with the original function:** $$y = \sqrt[4]{x} - 2$$ 4. **Isolate the fourth root term:** $$y + 2 = \sqrt[4]{x}$$ 5. **Rewrite the fourth root as an exponent:** $$y + 2 = x^{\frac{1}{4}}$$ 6. **Raise both sides to the power of 4 to cancel the root:** $$\left(y + 2\right)^4 = \left(x^{\frac{1}{4}}\right)^4$$ 7. **Simplify the right side:** $$\left(y + 2\right)^4 = x$$ 8. **Interchange $$x$$ and $$y$$ to write the inverse function:** $$y = \left(x + 2\right)^4$$ **Final answer:** $$f^{-1}(x) = \left(x + 2\right)^4$$