Question

select the correct answer from each drop - down menu.
the function ( f(x)=x^{3} ) has been transformed, resulting in function ( m ).
( m(x)=\frac{1}{3} x^{3}+6 )
as ( x ) approaches positive infinity, ( m(x) ) approaches
as ( x ) approaches negative infinity, ( m(x) ) approaches

Explanation:

Step1: Analyze the leading term

The leading term of \(m(x)=\frac{1}{3}x^{3}+6\) is \(\frac{1}{3}x^{3}\). As \(x\to+\infty\), \(x^{3}\to+\infty\), so \(\frac{1}{3}x^{3}\to+\infty\). And \(\frac{1}{3}x^{3}+6\approx\frac{1}{3}x^{3}\) (since the constant \(6\) is negligible compared to \(\frac{1}{3}x^{3}\) as \(x\) gets very large).

Step2: Analyze the behavior as \(x\to-\infty\)

As \(x\to-\infty\), \(x^{3}\to-\infty\), so \(\frac{1}{3}x^{3}\to-\infty\). And \(\frac{1}{3}x^{3}+6\approx\frac{1}{3}x^{3}\) (since the constant \(6\) is negligible compared to \(\frac{1}{3}x^{3}\) as \(x\) gets very large in the negative direction).

Answer:

As \(x\) approaches positive infinity, \(m(x)\) approaches \(+\infty\). As \(x\) approaches negative infinity, \(m(x)\) approaches \(-\infty\).