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.$g\left(x\right)=x^3-1$

Short answer

The graph of the function$g\left(x\right)=x^3-1$is given as:

The domain of the function is $\left(-\infty ,\infty \right)$and the range is$\left(-\infty ,\infty \right)$

Step-by-step solution

Step 1. given data

The given function is$g\left(x\right)=x^3-1$

Step 2. Graph the parent function. 

The parent function of the given function is$f\left(x\right)=x^3$

Plot the graph of the function$f\left(x\right)=x^3$

Step 3. Graph of the function 

Graph of function $g\left(x\right)=x^3-1$can be obtained by shifting downward the graph of the function $f\left(x\right)=x^3$by one unit

The graph of the given function is given as:

Step 4.  domain and range of the function

From the graph, we infer that

The domain of the function is$\left(-\infty ,\infty \right)$

The range of the function is$\left(-\infty ,\infty \right)$

Step 5. Verification

plot the graph of the function$g\left(x\right)=x^3-1$using a graphing utility

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