$39-50$ Find the domain and sketch the graph of the function.$f(x)=\{\begin{array}{ll}x+2& \text{ if }x<0\\ 1-x& \text{ if }x⩾0\end{array}$

Key Concepts: Piecewise Functions, Domain, Graphing Functions Explanation: The given function is a piecewise function, which means that it is defined by multiple sub-functions defined on different intervals of the real number line. In this case, there are two sub-functions: $f_1(x)=x+2$ for $x<0$ and $f_2(x)=1-x$ for $x\ge 0$. To graph the function $f(x)$, we need to graph both sub-functions on their respective domains and combine them to form the overall graph. Step by Step Solution: Step 1. To find the domain of $f(x)$, we need to determine the values of $x$ for which each sub-function is defined. The first sub-function is defined for $x<0$, which means that $f(x)$ is defined for all $x$ less than $0$. The second sub-function is defined for $x\ge 0$, which means that $f(x)$ is defined for all $x$ greater than or equal to $0$. Therefore, the domain of $f(x)$ is $(-\infty ,0)\cup [0,\infty )$. Step 2. To sketch the graph of $f(x)$, we can start by graphing each sub-function individually. The graph of $f_1(x)=x+2$ is a straight line passing through the point $(-2,0)$ with a slope of $1$, since $f_1(0)=2$. The graph of $f_2(x)=1-x$ is also a straight line passing through the point $(0,1)$ with a slope of $-1$. Step 3. To combine the two sub-functions and obtain the graph of $f(x)$, we need to plot the points where the two sub-functions meet, which is at $(0,1)$. We can see that the graph of $f(x)$ consists of two straight lines connected at the point $(0,1)$: the line $y=x+2$ for $x<0$, and the line $y=1-x$ for $x\ge 0$. Step 4. The final answer is shown in the attached graph. Final Answer: The domain of the function $f(x)$ is $(-\infty ,0)\cup [0,\infty )$. \n The graph of $f(x)$ consists of two straight lines connected at the point $(0,1)$: the line $y=x+2$ for $x<0$, and the line $y=1-x$ for $x\ge 0$. \n See attached graph for visual representation.