Question

Graph each function. Identify the domain and range.

$g\left(x\right)=\{\begin{array}{l}-1ifx\le -2\\ xif-2<x<2\\ -x+1ifx\ge 2\end{array}$

Short answer

The domain of the function is all real numbers and range is $\left\{y\right|y<2\}$.

The graph of the function is shown below:

Step-by-step solution

Step-1 – Domain and range of the function.

The domain of the function is the set of all input values and range of the function is the set of all output values.

Step-2 – Draw the graph.

The given function is$g\left(x\right)=\{\begin{array}{l}-1ifx\le -2\\ xif-2<x<2\\ -x+1ifx\ge 2\end{array}$

The graph of the equation $g\left(x\right)=-1$ is constant function for $x\le -2$.

Here, $-2$ satisfies the inequality $x\le -2$. Therefore, end with a closed circle at $x=-2$.

The graph of the equation $g\left(x\right)=x$ is linear equation for $-2<x<2$.

Here, $-2 and 2$ does not satisfies the inequality $-2<x<2$. Therefore, begin and closed with a closed circle at $x=-2 and x=2$.

The graph of the equation $g\left(x\right)=-x+1$ is linear equation for $x\ge 2$.

Here, 2 satisfies the inequality $x\ge 2$. Therefore, start with a closed circle at $x=2$.

The graph of the function is shown below:

Step-3 – Determine the domain and range.

Here, it can be observed that the domain of the function is the set of all real values because the function can take any value of x.

Also, the range is $\left\{y\right|y<2\}$.