Question: Use the graph of the function.
21. Identify the domain and range of the function.
22. Identify the x- and y-intercepts of the function.
23. Determine over what interval the function is positive or negative.
24. Determine over what interval the function is increasing or decreasing.
21. The domain is ☐.
(Type your answer in interval notation. Use integers or simplified fractions for any numbers in the express
1. **Problem Statement:** We are given a piecewise linear function with points at $(-5,0)$, $(-1,2)$, and $(5,-1)$. We need to find the domain, range, intercepts, intervals where the function is positive/negative, and intervals where it is increasing/decreasing.
2. **Domain:** The domain is the set of all $x$-values for which the function is defined. From the graph, the function starts at $x=-5$ and ends at $x=5$. So, the domain is $$[-5,5].$$
3. **Range:** The range is the set of all $y$-values the function takes. The minimum $y$-value is $-1$ at $x=5$, and the maximum $y$-value is $2$ at $x=-1$. So, the range is $$[-1,2].$$
4. **Intercepts:**
- **x-intercepts:** Points where $y=0$. From the graph, these are at $x=-5$ and $x=0$ (since the function crosses the x-axis between $-1$ and $5$, we find the exact intercept by linear interpolation).
To find the x-intercept between $(-1,2)$ and $(5,-1)$:
The slope is $$m=\frac{-1-2}{5-(-1)}=\frac{-3}{6}=-\frac{1}{2}.$$
Equation of the line segment:
$$y-2 = -\frac{1}{2}(x+1)$$
Set $y=0$ to find x-intercept:
$$0-2 = -\frac{1}{2}(x+1) \Rightarrow -2 = -\frac{1}{2}(x+1)$$
Multiply both sides by $-2$:
$$4 = x+1 \Rightarrow x=3.$$
So, x-intercepts are at $$x=-5$$ and $$x=3.$$
- **y-intercept:** Point where $x=0$. Using the same line segment between $(-1,2)$ and $(5,-1)$:
$$y-2 = -\frac{1}{2}(0+1) = -\frac{1}{2}$$
$$y = 2 - \frac{1}{2} = \frac{3}{2} = 1.5.$$
So, the y-intercept is at $$(0, 1.5).$$
5. **Intervals where the function is positive or negative:**
- The function is positive where $y>0$.
From $x=-5$ to $x=0$, $y$ is $\geq 0$ (since at $x=-5$, $y=0$ and at $x=0$, $y=1.5$).
From $x=0$ to $x=3$, $y$ decreases from $1.5$ to $0$, so still positive.
From $x=3$ to $x=5$, $y$ goes from $0$ to $-1$, so negative.
Therefore, positive interval is $$[-5,3)$$ and negative interval is $$(3,5].$$
6. **Intervals where the function is increasing or decreasing:**
- From $x=-5$ to $x=-1$, the function goes from $0$ to $2$, so it is increasing.
- From $x=-1$ to $x=5$, the function goes from $2$ to $-1$, so it is decreasing.
Therefore, increasing interval is $$[-5,-1]$$ and decreasing interval is $$[-1,5].$$