Question

Find the domain and sketch the graph of the functions\(f\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{x + 2}},{\bf{x}} < {\bf{0}}\\{\bf{1 - x}},{\bf{x}} \ge {\bf{0}}\end{array} \right.\).

Short answer

The domain of the function\(f\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{x + 2}},{\bf{x}} < {\bf{0}}\\{\bf{1 - x}},{\bf{x}} \ge {\bf{0}}\end{array} \right.\)is\(\left( { - \infty , + \infty } \right)\)and the graph of the function is given in figure (1).

Step-by-step solution

Determine the domain of the function

The given piecewise function is\({\rm{f}}\left( {\rm{x}} \right){\rm{ = }}\left\{ \begin{array}{l}{\rm{x + 2,x < 0}}\\{\rm{1 - x,x}} \ge {\rm{0}}\end{array} \right.\).

The function\(x + 2\)is defined for\(x < 0\), and the function\(1 - x\)is defined for\(x \ge 0\).

Therefore, the function is defined for\(\left( { - \infty ,\infty } \right)\).

Thus, the domain of the function \({\rm{f}}\left( {\rm{x}} \right){\rm{ = }}\left\{ \begin{array}{l}{\rm{x + 2,x < 0}}\\{\rm{1 - x,x}} \ge {\rm{0}}\end{array} \right.\)is\(\left( { - \infty ,\infty } \right)\).

Sketch the graph of the function

To draw the given piecewise function, draw\(x + 2\)for\(x < 0\), and draw\(1 - x\)for\(x \ge 0\).

Therefore, the graph of the function is given below in which the blue line shows the line for \(x < 0\) and the red line for\(x \ge 0\).

Figure (1)

Therefore, the graph of the function is given in figure (1).