Q. 99 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 97-100,graph each function,Based on the graph,state the domain and the range and find any intercepts.

$f\left(x\right)=\left\{\begin{array}{l}-e^x if x<0\\ -e^{-x } if x\ge 0\end{array}\right.$

Step-by-Step Solution

Answer

Graph of the given function

The domain of the graph is the set of all real numbers or the interval $(-\infty ,\infty )$ . Since $f\left(x\right)$  is negative for any value of x , the graph of $f\left(x\right)$  will never cross the x- axis. So, the range is the interval $[1,0)$ . The only intercept is $(0,-1)$ . 

Step 1.Given information

The given function $f\left(x\right)=\left\{\begin{array}{l}-e^x if x<0\\ -e^{-x } if x\ge 0\end{array}\right.$

Step 2.Draw graph for f ( x ) = - e x

The graph of $f\left(x\right)=-e^x$ for x<0 consits of onlybthe negtive values of x.

Step 3.Draw graph for f ( x ) = - e - x

The graph of $f\left(x\right)=-e^{-x}$for x $\ge 0$ consits of only the positive values of x including 0.

Step 4.Draw graph for both f ( x ) = - e x &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; i f &#160; x &#60; 0 - e - x &#160; &#160; &#160; &#160; &#160; &#160; &#160; i f &#160; &#160; x &#8805; 0

The graph of $f\left(x\right)=\left\{\begin{array}{l}-e^x if x<0\\ -e^{-x } if x\ge 0\end{array}\right.$will consits of both the graphs of $f\left(x\right)=e^x$ for x >0 and $f\left(x\right)=-e^{-x}$ for x$\ge 0$

From the graph, we can say that the domain of the graph is the set of all real numbers or the interval $(-\infty ,\infty )$ . Since $f\left(x\right)$ is negative for any value of x , the graph of $f\left(x\right)$ will never cross the x- axis. So, the range is the interval $[-1,0)$ . The only intercept is $(0,-1)$.