Q. 100 - 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

The graph of the given function 

The domain of the given function  $(-\infty ,\infty )$   The range of the given function  $(-\infty ,-1)$

The graph never touches the x-axis, there is no x-intercept. 

y- intercepts $(0,-1)$

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.Finding the x and y values for f ( x ) = - e - x

Let us begin with the graph of $f\left(x\right)$ or $y=-e^{-x}$ Since this function is defined for the value of x less than zero, select some values for x that are less than 0 and find the corresponding values of y

Create a table using the values. 

x	y = -e-x
-1	-2.178
-$\frac{3}{2}$	-4.482
-2	-7.389

Step 3.Finding the x and y values for f ( x ) = - e x

Similarly, for the function $y=-e^x$, select some values for x that are greater than or equal to 0 .

x	y = -ex
0	-1
1	-2.718
2	-7.389

Step 4.Plot the points for the two functions. Connect each set of points using a smooth curve.

The domain of a function is the set of all x-values for which the function is defined. We can see that $f\left(x\right)$ is defined for all real numbers. The domain of $f\left(x\right)$ is the interval $(-\infty ,\infty )$.

The range is the set of values of the function. From the graph, it is possible to say that the range of $f\left(x\right)$ is the interval $(-\infty ,-1)$.
The y-intercept is $(0,-1)$ . Since the graph never touches the x-axis, there is no x-intercept.