1. **State the problem:** Determine the domain and range of the function $f(x)$ based on the given graph description. 2. **Recall definitions:** - The **domain** of a function is the set of all possible input values ($x$) for which the function is defined. - The **range** of a function is the set of all possible output values ($f(x)$) the function can take. 3. **Analyze the graph description:** - The curve starts at $(-4,5)$ and decreases to $(0,0)$, so the function is defined for $x$ from $-4$ to $0$. - There is a point at $(1,1)$, so $x=1$ is in the domain. - There is a horizontal segment from $(1,3)$ to $(2,3)$, so the function is defined for $x$ in $[1,2]$ with $f(x)=3$. - There is a line segment from $(2,3)$ to $(3,4)$, so the function is defined for $x$ in $[2,3]$ with $f(x)$ increasing from $3$ to $4$. 4. **Determine the domain:** - From the above, the domain includes $[-4,0]$, the point $x=1$, and the interval $[1,3]$. - Since $x=1$ is included and the segment from $1$ to $3$ is continuous, the domain is $[-4,0] \cup [1,3]$. 5. **Determine the range:** - On $[-4,0]$, $f(x)$ goes from $5$ down to $0$, so range includes $[0,5]$. - At $x=1$, $f(1)=1$ which is already within $[0,5]$. - On $[1,2]$, $f(x)=3$ (horizontal segment), so $3$ is in the range. - On $[2,3]$, $f(x)$ goes from $3$ to $4$, so range includes $[3,4]$. - Combining all, the range is $[0,5]$. 6. **Final answers:** - Domain: $$[-4,0] \cup [1,3]$$ - Range: $$[0,5]$$