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.

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

Short 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$\left(-\infty \hspace{0.17em},\hspace{0.17em}0\right)\cup \left(0,\hspace{0.17em}\infty \hspace{0.17em}\right)$

Step-by-step solution

Step 1. Given information  

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

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 given function    

We know that for,$y=af\left(x\right)$, the y-coordinates of the original graph is multiplied $a$.

So, multiplying $\frac{1}{2}$to $f\left(x\right)$, the graph we get:

Step 4. Find the domain and range     

From the graph, we infer that

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

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

Step 5. 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.