Problem

Suppose $f(x)=2.2 \cdot x^{4}$ and $g(x)=-0.9 \cdot x^{3}$. Which of the following statements is not true?
As $x \rightarrow-\infty, f(x) \rightarrow \infty$.
As $x \rightarrow \infty, g(x) \rightarrow-\infty$
As $x \rightarrow \infty, f(x) \rightarrow \infty$.
As $x \rightarrow-\infty, g(x) \rightarrow-\infty$
As $x \rightarrow-\infty, g(x) \rightarrow \infty$.

Answer

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Answer

\(\boxed{\text{The statement that is not true is 'As } x \rightarrow-\infty, g(x) \rightarrow-\infty\text{'. The correct behavior of } g(x) \text{ as } x \text{ approaches negative infinity is that } g(x) \text{ approaches positive infinity.}}\)

Steps

Step 1 :Given the functions \(f(x)=2.2 \cdot x^{4}\) and \(g(x)=-0.9 \cdot x^{3}\), we are asked to determine which of the following statements is not true.

Step 2 :For the function \(f(x)\), since the power of \(x\) is even (4), the function will approach positive infinity as \(x\) approaches both positive and negative infinity. This is because any real number raised to an even power is always positive.

Step 3 :For the function \(g(x)\), since the power of \(x\) is odd (3), the function will approach positive infinity as \(x\) approaches negative infinity and negative infinity as \(x\) approaches positive infinity. This is because any real number raised to an odd power maintains its sign.

Step 4 :From the above analysis, we can see that the statement 'As \(x \rightarrow-\infty, g(x) \rightarrow-\infty\)' is not true. The correct behavior of \(g(x)\) as \(x\) approaches negative infinity is that \(g(x)\) approaches positive infinity.

Step 5 :\(\boxed{\text{The statement that is not true is 'As } x \rightarrow-\infty, g(x) \rightarrow-\infty\text{'. The correct behavior of } g(x) \text{ as } x \text{ approaches negative infinity is that } g(x) \text{ approaches positive infinity.}}\)

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