1. **State the problem:** Determine which of the given functions represents a nonlinear function of $x$. 2. **Recall the definition:** A function is linear if it can be written in the form $$y = mx + b$$ where $m$ and $b$ are constants, and the variable $x$ is to the first power only, with no products or divisions involving $x$. 3. **Analyze each function:** - For $$y = \frac{x + 3}{4} - 2x$$, simplify: $$y = \frac{x}{4} + \frac{3}{4} - 2x = \frac{x}{4} - 2x + \frac{3}{4}$$ Combine like terms: $$y = \frac{x}{4} - \frac{8x}{4} + \frac{3}{4} = -\frac{7x}{4} + \frac{3}{4}$$ This is linear because it is in the form $y = mx + b$. - For $$y = \frac{2}{x + 3}$$, the variable $x$ is in the denominator, which is not allowed in linear functions. This is nonlinear. - For $$y = \frac{x}{4} + 5$$, this is linear since it is $y = mx + b$ with $m=\frac{1}{4}$ and $b=5$. - For $$y = 10 + x$$, this is linear with $m=1$ and $b=10$. 4. **Conclusion:** The only nonlinear function is $$y = \frac{2}{x + 3}$$. **Final answer:** $$y = \frac{2}{x + 3}$$ is nonlinear.