1. **State the problem:** Find the inverse of the function $f(x) = 2x^2 - 10x + 14$. 2. **Rewrite the function:** Let $y = 2x^2 - 10x + 14$. 3. **Complete the square to simplify:** $$y = 2x^2 - 10x + 14 = 2(x^2 - 5x) + 14$$ Complete the square inside the parentheses: $$x^2 - 5x = x^2 - 5x + \left(\frac{-5}{2}\right)^2 - \left(\frac{-5}{2}\right)^2 = (x - \frac{5}{2})^2 - \frac{25}{4}$$ 4. **Substitute back:** $$y = 2\left((x - \frac{5}{2})^2 - \frac{25}{4}\right) + 14 = 2(x - \frac{5}{2})^2 - \frac{50}{4} + 14 = 2(x - \frac{5}{2})^2 - 12.5 + 14$$ 5. **Simplify constants:** $$y = 2(x - \frac{5}{2})^2 + 1.5$$ 6. **Solve for $x$ in terms of $y$:** $$y - 1.5 = 2(x - \frac{5}{2})^2$$ $$\frac{y - 1.5}{2} = (x - \frac{5}{2})^2$$ 7. **Take the square root:** $$x - \frac{5}{2} = \pm \sqrt{\frac{y - 1.5}{2}}$$ 8. **Solve for $x$:** $$x = \frac{5}{2} \pm \sqrt{\frac{y - 1.5}{2}}$$ 9. **Write the inverse function:** Since the original function is quadratic and not one-to-one over all real numbers, the inverse is defined piecewise: $$f^{-1}(y) = \frac{5}{2} + \sqrt{\frac{y - 1.5}{2}} \quad \text{or} \quad f^{-1}(y) = \frac{5}{2} - \sqrt{\frac{y - 1.5}{2}}$$ 10. **Domain restriction:** The inverse exists only for $y \geq 1.5$ because the square root requires non-negative input. **Final answer:** $$f^{-1}(y) = \frac{5}{2} \pm \sqrt{\frac{y - 1.5}{2}}$$