Piecewise Function 0D144F

1. **State the problem:** We have a piecewise function: $$f(x) = \begin{cases} x^2 & \text{if } 0 < x \leq 5 \\ 5x & \text{if } 5 < x < 8 \\ 40 & \text{if } x \geq 8 \end{cases}$$ We need to find the domain, intercepts, graph it, and find the range. 2. **Find the domain:** The domain is all $x$ values for which $f(x)$ is defined. - For $x^2$, domain is $0 < x \leq 5$ - For $5x$, domain is $5 < x < 8$ - For $40$, domain is $x \geq 8$ Combining these, the domain is: $$\boxed{(0, \infty)}$$ 3. **Locate intercepts:** - **x-intercepts:** Solve $f(x) = 0$ For $x^2 = 0$, $x=0$ but $x=0$ is not in domain (since $0 < x$), so no intercept here. For $5x=0$, $x=0$ but $0$ not in domain for this piece. For $40=0$, no solution. So no x-intercepts. - **y-intercept:** $f(0)$ is undefined (since domain excludes $0$), so no y-intercept. 4. **Graph the function:** - From $0 < x \leq 5$, graph $y = x^2$ starting just right of $0$ (open circle at $x=0$) up to and including $(5, 25)$ (closed circle). - From $5 < x < 8$, graph $y = 5x$ starting just right of $5$ (open circle at $(5, 25)$) to just left of $8$ (open circle at $(8, 40)$). - For $x \geq 8$, graph horizontal line $y=40$ starting at $(8, 40)$ (closed circle) extending right. 5. **Find the range:** - From $0 < x \leq 5$, $y = x^2$ ranges from just above $0$ to $25$. - From $5 < x < 8$, $y = 5x$ ranges from just above $25$ to just below $40$. - For $x \geq 8$, $y=40$ constant. Combining, the range is: $$\boxed{(0, 40]}$$ **Final answers:** - Domain: $(0, \infty)$ - No intercepts - Range: $(0, 40]$