1. **State the problem:** We are given the function $$y = \frac{4}{x} + \sqrt{x} + 0.2 - 5x$$ and the value $$x = \frac{4}{5}$$. We need to find the value of $$y$$ at this $$x$$. 2. **Substitute the value of $$x$$ into the function:** $$y = \frac{4}{\frac{4}{5}} + \sqrt{\frac{4}{5}} + 0.2 - 5 \times \frac{4}{5}$$ 3. **Simplify each term:** - For the first term: $$\frac{4}{\frac{4}{5}} = 4 \times \frac{5}{4} = \cancel{4} \times \frac{5}{\cancel{4}} = 5$$ - For the square root term: $$\sqrt{\frac{4}{5}} = \frac{\sqrt{4}}{\sqrt{5}} = \frac{2}{\sqrt{5}}$$ - For the linear term: $$5 \times \frac{4}{5} = \cancel{5} \times \frac{4}{\cancel{5}} = 4$$ 4. **Rewrite the expression with simplified terms:** $$y = 5 + \frac{2}{\sqrt{5}} + 0.2 - 4$$ 5. **Combine like terms:** $$y = (5 - 4 + 0.2) + \frac{2}{\sqrt{5}} = 1.2 + \frac{2}{\sqrt{5}}$$ 6. **Rationalize the denominator of the fraction:** $$\frac{2}{\sqrt{5}} = \frac{2}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$$ 7. **Final expression for $$y$$:** $$y = 1.2 + \frac{2\sqrt{5}}{5}$$ This is the exact value of $$y$$ at $$x = \frac{4}{5}$$. **Approximate numerical value:** Since $$\sqrt{5} \approx 2.236$$, $$\frac{2 \times 2.236}{5} = \frac{4.472}{5} = 0.8944$$ Therefore, $$y \approx 1.2 + 0.8944 = 2.0944$$ **Answer:** $$y = 1.2 + \frac{2\sqrt{5}}{5} \approx 2.0944$$