Chapter 3.7, Problem 3GP

To determine

To identify: The domain and range of a function

  $\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{l}2\mathrm{x}+1;\text{if}\mathrm{x}>0\\ 3;\text{if}\mathrm{x}\le 0\end{array}$

Answer to Problem 3GP

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

Range = $\left(1,\mathrm{\infty }\right)$

Explanation of Solution

Given information:

The function,

  $\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{l}2\mathrm{x}+1;\text{if}\mathrm{x}>0\\ 3;\text{if}\mathrm{x}\le 0\end{array}$

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

  

From the graph,

When

  $\mathrm{f}\left(\mathrm{x}\right)=\{2\mathrm{x}+1;\text{if}\mathrm{x}>0$ , the domain of function is $\left(0,\mathrm{\infty }\right)$ .

And

When

  $\mathrm{f}\left(\mathrm{x}\right)=\{3;\text{if}\mathrm{x}\le 0$ , the domain of function is $\left(-\mathrm{\infty },0\right]$ .

So, the combine domain of function

  $\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{l}2\mathrm{x}+1;\text{if}\mathrm{x}>0\\ 3;\text{if}\mathrm{x}\le 0\end{array}$ is $\left(-\mathrm{\infty },\mathrm{\infty }\right)$ .

The range of function $\mathrm{f}\left(\mathrm{x}\right)=\{2\mathrm{x}+1;\text{if}\mathrm{x}>0$ is $\left(1,\mathrm{\infty }\right)$ .

The range of function $\mathrm{f}\left(\mathrm{x}\right)=\{3;\text{if}\mathrm{x}\le 0$ is $\mathrm{f}\left(\mathrm{x}\right)=3$ .

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

  $\left(1,\mathrm{\infty }\right)$ .

Conclusion:

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

Range = $\left(1,\mathrm{\infty }\right)$