1. **State the problem:** Find the inverse function of $f(x) = 7x^3 - \frac{2}{20}$ where $x \in \mathbb{R}$. 2. **Rewrite the function:** Simplify the constant term: $f(x) = 7x^3 - \frac{1}{10}$. 3. **Set $y = f(x)$:** $$y = 7x^3 - \frac{1}{10}$$ 4. **Solve for $x$ in terms of $y$:** Add $\frac{1}{10}$ to both sides: $$y + \frac{1}{10} = 7x^3$$ Divide both sides by 7: $$\frac{y + \frac{1}{10}}{7} = x^3$$ Take the cube root of both sides: $$x = \sqrt[3]{\frac{y + \frac{1}{10}}{7}}$$ 5. **Write the inverse function:** $$f^{-1}(y) = \sqrt[3]{\frac{y + \frac{1}{10}}{7}}$$ 6. **Interpretation:** The inverse function takes an output $y$ of the original function and returns the corresponding input $x$. Since the original function is a cubic polynomial with domain and range $\mathbb{R}$, the inverse exists and is also defined for all real numbers.