Question

1. target e

determine if the degree of the function of the graph is even or odd. explain how you know this based on the end behavior.

Explanation:

Step1: Recall End Behavior Rules

For a polynomial function \( f(x)=a_nx^n + a_{n - 1}x^{n - 1}+\dots+a_1x + a_0\), the end - behavior is determined by the leading term \(a_nx^n\).

• If \(n\) is even:
• If \(a_n>0\), as \(x

ightarrow+\infty\), \(f(x)
ightarrow+\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\).

• If \(a_n < 0\), as \(x

ightarrow+\infty\), \(f(x)
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\).

• If \(n\) is odd:
• If \(a_n>0\), as \(x

ightarrow+\infty\), \(f(x)
ightarrow+\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow-\infty\).

• If \(a_n < 0\), as \(x

ightarrow+\infty\), \(f(x)
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(f(x)
ightarrow+\infty\).

Step2: Analyze the Given Graph's End Behavior

From the graph, as \(x
ightarrow+\infty\), the function \(f(x)\) goes to \(+\infty\) (assuming the right - hand end of the upper curve and the right - hand end of the lower curve, looking at the direction of the arrows). As \(x
ightarrow-\infty\), the function \(f(x)\) also goes to \(+\infty\) (the left - hand end of the upper curve).

This type of end - behavior (both ends going to the same infinity, either both \(+\infty\) or both \(-\infty\)) is characteristic of a polynomial with an even degree. Because when the degree \(n\) is even, the leading term \(a_nx^n\) will have the same sign for \(x
ightarrow+\infty\) and \(x
ightarrow-\infty\) (since \(x^n\) is positive when \(n\) is even, regardless of the sign of \(x\)).

Answer:

The degree of the function is even. This is because the end - behavior of the graph (both ends of the graph going to \(+\infty\) as \(x
ightarrow\pm\infty\)) is consistent with the end - behavior of a polynomial function with an even degree (for an even - degree polynomial, as \(x
ightarrow+\infty\) and \(x
ightarrow-\infty\), the function values have the same sign and tend to the same infinity, either both \(+\infty\) or both \(-\infty\)).