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Describe the end behavior of the function below.

(2 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
As x approaches −∞, f(x) approaches
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As x approaches ∞, f(x) approaches

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1 answer

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 \( -\infty \):

    • 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 \( \infty \):

    • 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 \( -\infty \), \( f(x) \) approaches \( +\infty \).
    • As \( x \) approaches \( \infty \), \( f(x) \) approaches \( +\infty \).

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

    • As \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \).
    • As \( x \) approaches \( \infty \), \( f(x) \) approaches \( +\infty \).

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

    • As \( x \) approaches \( -\infty \), \( f(x) \) approaches \( -\infty \).
    • As \( x \) approaches \( \infty \), \( f(x) \) approaches \( -\infty \).

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

    • As \( x \) approaches \( -\infty \), \( f(x) \) approaches \( +\infty \).
    • As \( x \) approaches \( \infty \), \( f(x) \) approaches \( -\infty \).

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.