1. **Problem Statement:** Given the function $$y = 10x^2 - 4x + 3$$, find its inverse function. 2. **Understanding the Problem:** The inverse function, denoted as $$x = f^{-1}(y)$$, reverses the roles of $$x$$ and $$y$$. To find the inverse, we solve for $$x$$ in terms of $$y$$. 3. **Step 1: Write the equation:** $$y = 10x^2 - 4x + 3$$ 4. **Step 2: Rearrange to standard quadratic form:** $$10x^2 - 4x + (3 - y) = 0$$ 5. **Step 3: Use the quadratic formula to solve for $$x$$:** The quadratic formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=10$$, $$b=-4$$, and $$c=3 - y$$. 6. **Step 4: Substitute values:** $$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \times 10 \times (3 - y)}}{2 \times 10} = \frac{4 \pm \sqrt{16 - 40(3 - y)}}{20}$$ 7. **Step 5: Simplify under the square root:** $$16 - 40(3 - y) = 16 - 120 + 40y = 40y - 104$$ 8. **Step 6: Write the inverse function:** $$x = \frac{4 \pm \sqrt{40y - 104}}{20}$$ 9. **Step 7: Express inverse function explicitly:** $$f^{-1}(y) = \frac{4 \pm \sqrt{40y - 104}}{20}$$ 10. **Note:** Since the original function is quadratic (not one-to-one over all real numbers), the inverse is not a function unless the domain is restricted. The $$\pm$$ indicates two branches of the inverse. **Final answer:** $$f^{-1}(y) = \frac{4 \pm \sqrt{40y - 104}}{20}$$