Chapter 3.7, Problem 16PPS

To determine

To identify: The domain and range of a function

  $\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{l}\frac{1}{2}\mathrm{x}-1;\text{if}\mathrm{x}>3\\ -2\mathrm{x}+3;\text{if}\mathrm{x}\le 3\end{array}$

Answer to Problem 16PPS

Domain = $\left(-\mathrm{\infty },\mathrm{\infty }\right)$

Range = $\left[-3,\mathrm{\infty }\right)$

Explanation of Solution

Given information:

The function,

  $\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{l}\frac{1}{2}\mathrm{x}-1;\text{if}\mathrm{x}>3\\ -2\mathrm{x}+3;\text{if}\mathrm{x}\le 3\end{array}$

Sketch the graph of given function $\mathrm{f}\left(\mathrm{x}\right)$ .

  

From the graph,

When

  $\mathrm{f}\left(\mathrm{x}\right)=\{\frac{1}{2}\mathrm{x}-1;\text{if}\mathrm{x}>3$ , the domain of function is $\left(3,\mathrm{\infty }\right)$ .

And

When

  $\mathrm{f}\left(\mathrm{x}\right)=\{-2\mathrm{x}+3;\text{if}\mathrm{x}\le 3$ , the domain of function is

  $\left(-\mathrm{\infty },3\right]$ .

So, the combine domain of function

  $\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{l}\frac{1}{2}\mathrm{x}-1;\text{if}\mathrm{x}>3\\ -2\mathrm{x}+3;\text{if}\mathrm{x}\le 3\end{array}$

is $\left(-\mathrm{\infty },\mathrm{\infty }\right)$ .

The range of function $\mathrm{f}\left(\mathrm{x}\right)=\{\frac{1}{2}\mathrm{x}-1;\text{if}\mathrm{x}>3$ is $\left(\frac{1}{2},\mathrm{\infty }\right)$ .

The range of function $\mathrm{f}\left(\mathrm{x}\right)=\{-2\mathrm{x}+3;\text{if}\mathrm{x}\le 3$ is $\left[-3,\mathrm{\infty }\right)$ .

Then, the combine range of function $\mathrm{f}\left(\mathrm{x}\right)$ is

  $\left[-3,\mathrm{\infty }\right)$ .

Conclusion:

Domain = $\left(-\mathrm{\infty },\mathrm{\infty }\right)$

Range = $\left[-3,\mathrm{\infty }\right)$