Q. 56 - Precalculus Enhanced with Graphing Utilities

Question

Graph the function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function and show all stages. Be sure to show at least three key points. Find the domain and the range of each function. Verify your results using a graphing utility.  

$h\left(x\right)=\frac{1}{-x}+2$

Step-by-Step Solution

Answer

The graph of the given function is:   

The domain of the function is  $\left(-\infty \hspace{0.17em},\hspace{0.17em}0\right)\cup \left(0,\hspace{0.17em}\infty \hspace{0.17em}\right)$

The range of the function is  $\hspace{0.17em}\left(-\infty \hspace{0.17em},\hspace{0.17em}2\right)\cup \left(2,\hspace{0.17em}\infty \hspace{0.17em}\right)$

Step 1. Given information

The given function is $h\left(x\right)=\frac{1}{-x}+2$

Step 2. Graph the parent function

The parent function of the given function is $f\left(x\right)=\frac{1}{x}$.

The graph is given as:

Step 3. Graph the function g ( x ) = 1 - x

We know that, for the function $y=f(-x)$, the original graph is reflected over the y- axis 

The graph is given as:

Step 4. Graph the given function.

We know that, for the function $y=f\left(x\right)+k$, the original graph is shifted upwards by $k$ units.

Similarly, the graph $g\left(x\right)=\frac{1}{-x}$ is shifted upwards by $2$units.

The graph is given as:

Step 5. Find the domain and range

The domain of the given function is $\left(-\infty \hspace{0.17em},\hspace{0.17em}0\right)\cup \left(0,\hspace{0.17em}\infty \hspace{0.17em}\right)$

The range of the given function is $\hspace{0.17em}\left(-\infty \hspace{0.17em},\hspace{0.17em}2\right)\cup \left(2,\hspace{0.17em}\infty \hspace{0.17em}\right)$ 

Step 6. Verify the graph

The graph of the given function using graphing utility is given as : 

The graph is the same as we have drawn. Hence, it is correct.