Question

use a graphing utility to graph f and g in the same viewing rectangle. then use the zoom out feature to show f and g have identical end behavior.
f(x)=x^{3}-6x - 2
g(x)=x^{3}
graph f and g in the same viewing rectangle. use an initial viewing rectangle that matches one of the choices below. choose the correct graph below.
a.
b.
c.
d.

Explanation:

Step1: Analyze end - behavior of polynomials

For a polynomial function \(y = a_nx^n+\cdots+a_0\), the end - behavior is determined by the leading term \(a_nx^n\). For \(f(x)=x^3 - 6x - 2\) and \(g(x)=x^3\), the leading term for both is \(x^3\). When \(x\to+\infty\), \(y\to+\infty\) and when \(x\to-\infty\), \(y\to-\infty\) for both functions since the leading coefficient \(a = 1>0\) and the degree \(n = 3\) (odd).

Step2: Consider the difference between the functions

\(f(x)-g(x)=(x^3 - 6x - 2)-x^3=-6x - 2\), which is a linear function. This means \(f(x)\) is a vertical and horizontal shift (due to the linear term) of \(g(x)\). But the end - behavior remains the same.

Step3: Match with graphs

We need to find a graph where the two functions have the same end - behavior (both go to \(+\infty\) as \(x\to+\infty\) and to \(-\infty\) as \(x\to-\infty\)) and \(f(x)\) is a shifted version of \(g(x)\). Without seeing the actual details of the graphs in a more clear way, we know that the cubic function \(f(x)=x^3-6x - 2\) will have some local extrema due to the non - zero linear and constant terms, while \(g(x)=x^3\) has an inflection point at the origin. However, as we zoom out, the end - behavior is the key factor.

Answer:

(Here, since the actual visual details of the graphs A, B, C, D are not clear enough in the provided image, we cannot definitively choose one of the options. But the correct graph should show two cubic functions with the same end - behavior and \(f(x)\) being a shifted version of \(g(x)\))