Question

use a graphing utility to graph f and g in the same viewing rectangle. then use the zoom out feature to show that f and g have identical end behavior.
f(x)=x^3 - 6x - 2
g(x)=x^3
now use the zoom out feature for the graph found in the previous step. choose the correct graph below.
according to the graph shown in the previous step, do f and g have identical end behavior?

Explanation:

Step1: Recall end - behavior rule for polynomials

For a polynomial function \(y = a_nx^n+\cdots+a_0\), the end - behavior is determined by the leading term \(a_nx^n\). The function \(f(x)=x^{3}-6x - 2\) has a leading term \(x^{3}\), and \(g(x)=x^{3}\) also has a leading term \(x^{3}\).

Step2: Analyze end - behavior of cubic functions

For a cubic function \(y = ax^{3}\) with \(a = 1>0\), as \(x\to+\infty\), \(y\to+\infty\) and as \(x\to-\infty\), \(y\to-\infty\). Since both \(f(x)\) and \(g(x)\) have the same leading - term \(x^{3}\) (where \(a = 1\)), they have the same end - behavior.

Step3: Consider the graphs

When we zoom out on the graphs of \(f(x)=x^{3}-6x - 2\) and \(g(x)=x^{3}\), the difference in the non - leading terms (\(-6x-2\) in \(f(x)\)) becomes less significant, and the overall shape and end - behavior are dominated by the leading term \(x^{3}\).

Answer:

Yes