Question

question
the function ( f(x) ) is defined below. what is the end behavior of ( f(x) )?

( f(x)=-392 + 399x-7x^{3} )

answer attempt 1 out of 2

the degree of the polynomial is , and the leading coefficient is .

choose which pair of limits below represents the end behavior:

( lim _{x
ightarrow-infty} f(x)=infty, quad lim _{x
ightarrow infty} f(x)=infty )

( lim _{x
ightarrow-infty} f(x)=-infty, quad lim _{x
ightarrow infty} f(x)=infty )

( lim _{x
ightarrow-infty} f(x)=infty, quad lim _{x
ightarrow infty} f(x)=-infty )

( lim _{x
ightarrow-infty} f(x)=-infty, quad lim _{x
ightarrow infty} f(x)=-infty )

Explanation:

Step1: Find the degree of the polynomial

For a polynomial \(f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0\), the degree \(n\) is the highest power of \(x\). In \(f(x)=-392 + 399x-7x^{3}\), the highest power of \(x\) is \(3\).

Step2: Find the leading coefficient

The leading coefficient is the coefficient of the term with the highest - degree. For \(f(x)=-392 + 399x-7x^{3}\), the term with the highest degree is \(-7x^{3}\), and its coefficient is \(-7\).

Step3: Determine the end - behavior

For a polynomial \(y = ax^n\) (where \(a\) is the leading coefficient and \(n\) is the degree):

• When \(n\) is odd:
• If \(a<0\), as \(x\to\infty\), \(y = ax^n\to-\infty\) (since \(x^n>0\) for \(x>0\) and \(a<0\)), and as \(x\to-\infty\), \(x^n<0\) (because \(n\) is odd) and \(y=ax^n>0\) (since \(a<0\) and \(x^n<0\), so \(a\times x^n>0\)).
• Here \(n = 3\) (odd) and \(a=-7<0\).
• \(\lim_{x\to-\infty}(-7x^{3})=\infty\) (because \(x^{3}\to-\infty\) as \(x\to-\infty\) and \(-7\times(-\infty)=\infty\)).
• \(\lim_{x\to\infty}(-7x^{3})=-\infty\) (because \(x^{3}\to\infty\) as \(x\to\infty\) and \(-7\times\infty=-\infty\)).
• For the polynomial \(f(x)=-392 + 399x-7x^{3}\), the end - behavior is dominated by the term \(-7x^{3}\). So \(\lim_{x\to-\infty}f(x)=\infty\) and \(\lim_{x\to\infty}f(x)=-\infty\).

Answer:

The degree of the polynomial is \(3\), and the leading coefficient is \(-7\). The correct pair of limits is \(\lim_{x\to-\infty}f(x)=\infty,\ \lim_{x\to\infty}f(x)=-\infty\) (the third option).