1. **State the problem:** We have a linear function $f(x)$ passing through points $(2,5)$ and $(-1,10)$. We need to find the slope of a line perpendicular to $f(x)$. 2. **Formula for slope:** The slope $m$ of a line through points $(x_1,y_1)$ and $(x_2,y_2)$ is given by: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 3. **Calculate the slope of $f(x)$:** $$m = \frac{10 - 5}{-1 - 2} = \frac{5}{-3} = -\frac{5}{3}$$ 4. **Slope of perpendicular line:** The slope of a line perpendicular to another is the negative reciprocal of the original slope. If original slope is $m$, perpendicular slope is: $$m_\perp = -\frac{1}{m}$$ 5. **Calculate perpendicular slope:** $$m_\perp = -\frac{1}{-\frac{5}{3}} = \frac{3}{5}$$ **Final answer:** The slope of a line perpendicular to $f(x)$ is $\boxed{\frac{3}{5}}$.