The Function \[$ G \$\] Is A Polynomial With The Following End Behavior:$\[ \lim _{x \rightarrow -\infty} G(x) = -\infty \quad \text{and} \quad \lim _{x \rightarrow \infty} G(x) = -\infty \\]Which Of The Following Could Be An

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Understanding the End Behavior of a Polynomial Function

The end behavior of a polynomial function is a crucial aspect of understanding its behavior as the input values approach positive or negative infinity. In this article, we will explore the end behavior of a polynomial function g(x) and determine which of the given options could be an expression for g(x).

The Given End Behavior

The given end behavior of the function g(x) is as follows:

{ \lim _{x \rightarrow -\infty} g(x) = -\infty \quad \text{and} \quad \lim _{x \rightarrow \infty} g(x) = -\infty \}

This means that as x approaches negative infinity, the value of g(x) also approaches negative infinity. Similarly, as x approaches positive infinity, the value of g(x) also approaches negative infinity.

Possible Forms of the Function g(x)

Given the end behavior of the function g(x), we can determine the possible forms of the function. Since the function is a polynomial, it can be expressed in the form:

g(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0

where a_n, a_(n-1), ..., a_1, a_0 are constants, and n is a positive integer.

Determining the Leading Coefficient

The leading coefficient of the polynomial function g(x) determines its end behavior. If the leading coefficient is positive, the function will approach positive infinity as x approaches positive infinity. If the leading coefficient is negative, the function will approach negative infinity as x approaches positive infinity.

In this case, since the function g(x) approaches negative infinity as x approaches positive infinity, the leading coefficient must be negative.

Possible Forms of the Leading Term

The leading term of the polynomial function g(x) is the term with the highest degree. Since the leading coefficient is negative, the leading term must be of the form:

-a_n x^n

where a_n is a positive constant.

Possible Forms of the Function g(x)

Given the leading term, we can determine the possible forms of the function g(x). Since the function is a polynomial, it can be expressed in the form:

g(x) = -a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0

where a_n, a_(n-1), ..., a_1, a_0 are constants, and n is a positive integer.

Example Forms of the Function g(x)

Here are some example forms of the function g(x):

  • g(x) = -x^3 + 2x^2 - 3x + 1
  • g(x) = -2x^4 + 3x^3 - 4x^2 + 5x - 6
  • g(x) = -x^5 + 2x^4 - 3x^3 + 4x^2 - 5x + 6

Conclusion

In conclusion, the function g(x) is a polynomial with the following end behavior:

{ \lim _{x \rightarrow -\infty} g(x) = -\infty \quad \text{and} \quad \lim _{x \rightarrow \infty} g(x) = -\infty \}

The possible forms of the function g(x) are determined by the leading coefficient and the leading term. The leading coefficient must be negative, and the leading term must be of the form:

-a_n x^n

where a_n is a positive constant.

The function g(x) can be expressed in the form:

g(x) = -a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0

where a_n, a_(n-1), ..., a_1, a_0 are constants, and n is a positive integer.

References

  • [1] "Polynomial Functions" by Math Open Reference
  • [2] "End Behavior of Polynomial Functions" by Purplemath

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Frequently Asked Questions

In this article, we will answer some frequently asked questions about the function g(x) and its end behavior.

Q: What is the end behavior of the function g(x)?

A: The end behavior of the function g(x) is such that as x approaches negative infinity, the value of g(x) also approaches negative infinity. Similarly, as x approaches positive infinity, the value of g(x) also approaches negative infinity.

Q: What is the leading coefficient of the function g(x)?

A: The leading coefficient of the function g(x) is negative.

Q: What is the leading term of the function g(x)?

A: The leading term of the function g(x) is of the form:

-a_n x^n

where a_n is a positive constant.

Q: How can I determine the possible forms of the function g(x)?

A: To determine the possible forms of the function g(x), you need to consider the leading coefficient and the leading term. The leading coefficient must be negative, and the leading term must be of the form:

-a_n x^n

where a_n is a positive constant.

Q: Can you give some examples of the function g(x)?

A: Yes, here are some examples of the function g(x):

  • g(x) = -x^3 + 2x^2 - 3x + 1
  • g(x) = -2x^4 + 3x^3 - 4x^2 + 5x - 6
  • g(x) = -x^5 + 2x^4 - 3x^3 + 4x^2 - 5x + 6

Q: How can I use the end behavior of the function g(x) to determine its graph?

A: To use the end behavior of the function g(x) to determine its graph, you need to consider the following:

  • As x approaches negative infinity, the value of g(x) approaches negative infinity.
  • As x approaches positive infinity, the value of g(x) approaches negative infinity.

This means that the graph of the function g(x) will have a horizontal asymptote at y = -∞.

Q: Can you give some tips for graphing the function g(x)?

A: Yes, here are some tips for graphing the function g(x):

  • Use a graphing calculator or software to graph the function g(x).
  • Consider the end behavior of the function g(x) to determine its horizontal asymptote.
  • Use the leading term to determine the behavior of the function g(x) near the origin.
  • Use the graphing calculator or software to zoom in and out of the graph to see the behavior of the function g(x) near the origin.

Conclusion

In conclusion, the function g(x) is a polynomial with the following end behavior:

{ \lim _{x \rightarrow -\infty} g(x) = -\infty \quad \text{and} \quad \lim _{x \rightarrow \infty} g(x) = -\infty \}

The possible forms of the function g(x) are determined by the leading coefficient and the leading term. The leading coefficient must be negative, and the leading term must be of the form:

-a_n x^n

where a_n is a positive constant.

We hope that this article has been helpful in answering your questions about the function g(x) and its end behavior.

References

  • [1] "Polynomial Functions" by Math Open Reference
  • [2] "End Behavior of Polynomial Functions" by Purplemath

Related Topics