Isolated Values 6D613F

1. **State the problem:** Determine if the piecewise function $$f(x) = \begin{cases} x^2 & \text{if } 0 < x \leq 5 \\ 5x & \text{if } 5 < x < 8 \\ 40 & \text{if } x \geq 8 \end{cases}$$ has any isolated values. 2. **Recall what isolated values mean:** An isolated value in the range of a function is a value that the function takes at a single point or over an interval that is not connected to other values in the range. 3. **Analyze each piece:** - For $0 < x \leq 5$, $f(x) = x^2$ is continuous and takes values from just above $0$ to $25$. - For $5 < x < 8$, $f(x) = 5x$ is continuous and takes values from just above $25$ to just below $40$. - For $x \geq 8$, $f(x) = 40$ is constant. 4. **Check for isolated values:** - The value $40$ is taken for all $x \geq 8$, so it is not isolated. - The function values form continuous intervals from just above $0$ to $40$ without gaps. 5. **Conclusion:** There are no isolated values in the function's range because the function covers the interval $(0,40]$ continuously and the constant value $40$ is taken on an interval, not a single point. **Final answer:** No, there are no isolated values in the function.