Describe the end behavior of the function below.

Question

(2 points)
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As x approaches −∞, f(x) approaches
.
As x approaches ∞, f(x) approaches

.

Answer 1

In order to describe the end behavior of a function, we typically need to analyze the function's leading term or its behavior as x approaches negative and positive infinity.

However, since you haven't provided the specific function in question, I'll explain the general approach for determining the end behavior:

1. As $x$ approaches $-\mathrm{\infty }$:

   • Determine if the leading term (the term with the highest power of $x$) approaches a specific value.
   • The behavior can be infinite (positive or negative) or could approach a constant.

2. As $x$ approaches $\mathrm{\infty }$:

   • Similar to the previous point, consider the leading term.
   • The end behavior can also be infinite (positive or negative) or could converge to a constant.

Without the function specifics, I can't provide the exact values to insert. Here are some general responses you could use based on the common types of polynomial functions:

• If the leading coefficient is positive and the degree of the polynomial is even, then:

  • As $x$ approaches $-\mathrm{\infty }$, $f(x)$ approaches $+\mathrm{\infty }$.
  • As $x$ approaches $\mathrm{\infty }$, $f(x)$ approaches $+\mathrm{\infty }$.

• If the leading coefficient is positive and the degree is odd, then:

  • As $x$ approaches $-\mathrm{\infty }$, $f(x)$ approaches $-\mathrm{\infty }$.
  • As $x$ approaches $\mathrm{\infty }$, $f(x)$ approaches $+\mathrm{\infty }$.

• If the leading coefficient is negative and the degree is even, then:

  • As $x$ approaches $-\mathrm{\infty }$, $f(x)$ approaches $-\mathrm{\infty }$.
  • As $x$ approaches $\mathrm{\infty }$, $f(x)$ approaches $-\mathrm{\infty }$.

• If the leading coefficient is negative and the degree is odd, then:

  • As $x$ approaches $-\mathrm{\infty }$, $f(x)$ approaches $+\mathrm{\infty }$.
  • As $x$ approaches $\mathrm{\infty }$, $f(x)$ approaches $-\mathrm{\infty }$.

Make sure to replace these placeholders with the specific asymptotic values or behaviors based on the function you are analyzing. If you provide the function itself, I can give you the exact end behavior values to insert.