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Chapter 2: Problem 98

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. Then use the \([\mathrm{ZOOMOUT}]\) feature to show that \(f\) and \(g\) have identical end behavior. $$f(x)=x^{3}-6 x+1, \quad g(x)=x^{3}$$

Short Answer

Expert verified
Both \(f(x) = x^3 - 6x + 1\) and \(g(x) = x^3\) have identical end behavior, tending towards positive and negative infinity as \(x\) tends towards positive and negative infinity respectively. This is shown by graphing both functions on the same viewing rectangle and using the ZOOMOUT feature to observe their behavior as \(x\) gets larger or smaller.

Step by step solution

01

Plot the functions

Start by plotting the two functions \(f(x) = x^3 - 6x + 1\) and \(g(x) = x^3\) in the same viewing rectangle. This will give an initial comparison of the two functions.
02

Use the ZOOMOUT feature

After having the two functions plotted, use the ZOOMOUT feature in the graphing tool. This helps to view the end behavior of the functions more clearly by seeing how they behave as \(x\) approaches positive and negative infinity.
03

Compare the end behaviors

Observe the two graphs as \(x\) increases and decreases without bound. The end behavior of a function refers to the behavior of the function values as \(x\) goes to positive or negative infinity. In this case, both \(f(x)\) and \(g(x)\) tend to positive and negative infinity as \(x\) tends to positive and negative infinity respectively. This is because the leading coefficient in both functions is positive. Since the functions share the same end behavior, we can conclude that they have an identical end behavior.

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