Question

Find the domain and sketch the graph of the functions\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{3 - }}\frac{{\bf{1}}}{{\bf{2}}}{\bf{x}},{\bf{x}} \le {\bf{2}}\\{\bf{2x - 5}},{\bf{x}} > {\bf{2}}\end{array} \right.\).

Short answer

The domain of the function\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{3 - }}\frac{{\bf{1}}}{{\bf{2}}}{\bf{x}},{\bf{x}} \le {\bf{2}}\\{\bf{2x - 5}},{\bf{x}} > {\bf{2}}\end{array} \right.\)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 function is\({\rm{f}}\left( {\rm{x}} \right){\rm{ = }}\left\{ \begin{array}{l}{\rm{3 - }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{x,x}} \le {\rm{2}}\\{\rm{2x - 5,x > 2}}\end{array} \right.\).

The function is a piecewise function in which\(3 - \frac{1}{2}x\)for\(x \le 2\)and\(2x - 5\)for\(x > 2\).

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

Thus, the domain of the function \({\rm{f}}\left( {\rm{x}} \right){\rm{ = }}\left\{ \begin{array}{l}{\rm{3 - }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{x,x}} \le {\rm{2}}\\{\rm{2x - 5,x > 2}}\end{array} \right.\)is\(\left( { - \infty ,\infty } \right)\).

Sketch the graph of the function

To draw the graph of the given function, draw\(3 - \frac{1}{2}x\)for\(x \le 2\)and\(2x - 5\)for\(x > 2\).

Therefore, the graph of the function is given below in which the blue line shows the line for \(x \le 2\) and the red line for\(x > 2\)

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