Question

Find the domain and sketch the graph of the functions\(f\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{ - 1}},{\bf{x}} \le {\bf{ - 1}}\\{\bf{3x + 2,}}\left| {\bf{x}} \right| < {\bf{1}}\\{\bf{7 - 2x,x}} \ge {\bf{1}}\end{array} \right.\).

Short answer

The domain of the function\(f\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{ - 1}},{\bf{x}} \le {\bf{ - 1}}\\{\bf{3x + 2,}}\left| {\bf{x}} \right| < {\bf{1}}\\{\bf{7 - 2x,x}} \ge {\bf{1}}\end{array} \right.\)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 domain of given function\({\rm{f}}\left( {\rm{x}} \right){\rm{ = }}\left\{ \begin{array}{l}{\rm{ - 1,x}} \le {\rm{ - 1}}\\{\rm{3x + 2,}}\left| {\rm{x}} \right|{\rm{ < 1}}\\{\rm{7 - 2x,x}} \ge {\rm{1}}\end{array} \right.\)is found by finding domain for each function separately.

When\(f\left( x \right) = - 1\)then domain will be\( - 1\)because it is a constant function.

When\(f\left( x \right) = 3x + 2\)with\(\left| x \right| < 1\), then domain of function is\(( - \infty ,1)\).

When\(f\left( x \right) = 7 - 2x\)with\(x \ge 1\), then the domain of function is\(\left( {1,\infty } \right)\).

The domain for the given combined piecewise function will be\(\left( { - \infty , + \infty } \right)\)

Therefore, the domain of the function \({\rm{f}}\left( {\rm{x}} \right){\rm{ = }}\left\{ \begin{array}{l}{\rm{ - 1,x}} \le {\rm{ - 1}}\\{\rm{3x + 2,}}\left| {\rm{x}} \right|{\rm{ < 1}}\\{\rm{7 - 2x,x}} \ge {\rm{1}}\end{array} \right.\)is\(\left( { - \infty , + \infty } \right)\).

Sketch the graph of the function

The graph for the \(f\left( x \right) = - 1\) is given below.

The graph for the function \(f\left( x \right) = 3x + 2\) is given below.

The graph for the function \(f\left( x \right) = 7 - 2x\) is given below:

The graph for the complete piecewise function is given in figure (1).

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