Question

the graph of ( y = f(x) ) is graphed below. what is the end behavior of ( f(x) )?

Explanation:

Step1: Analyze the left - hand end behavior

As \(x\to-\infty\), we observe the graph. The leading term of the polynomial (since it's a polynomial - like graph) determines the end - behavior. For a polynomial \(y = a_nx^n+\cdots+a_0\), when \(n\) is even and \(a_n>0\), as \(x\to-\infty\), \(y\to+\infty\). But looking at the graph, as \(x\to-\infty\), the graph goes up, so \(\lim_{x\to-\infty}f(x)=\infty\).

Step2: Analyze the right - hand end behavior

As \(x\to+\infty\), the leading term of the polynomial (assuming it's a polynomial function). When \(n\) is odd and \(a_n < 0\), as \(x\to+\infty\), \(y\to-\infty\). Looking at the graph, as \(x\to+\infty\), the graph goes down, so \(\lim_{x\to+\infty}f(x)=-\infty\).

Answer:

\(\lim_{x\to-\infty}f(x)=\infty\) and \(\lim_{x\to+\infty}f(x)=-\infty\)