Complete The Statements About The Key Features Of The Graph Of $f(x) = X^5 - 9x^3$.As $x$ Goes To Negative Infinity, $f(x$\] Goes To $\square$ Infinity, And As $x$ Goes To Positive Infinity, $f(x$\]

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Introduction

The graph of a function is a visual representation of the relationship between the input and output values of the function. In this article, we will focus on the key features of the graph of the function $f(x) = x^5 - 9x^3$. We will analyze the behavior of the function as $x$ approaches negative and positive infinity, and identify the key features of the graph.

Behavior of the Function as $x$ Approaches Negative Infinity

As $x$ approaches negative infinity, the value of $f(x)$ can be determined by analyzing the behavior of the function. Since the function is a polynomial function, we can use the properties of polynomial functions to determine the behavior of the function as $x$ approaches negative infinity.

The function $f(x) = x^5 - 9x^3$ is an odd-degree polynomial function, which means that it will have a horizontal asymptote at $y = 0$. This means that as $x$ approaches negative infinity, the value of $f(x)$ will approach negative infinity.

To determine the exact behavior of the function as $x$ approaches negative infinity, we can use the following inequality:

$$\lim_{x\to-\infty} f(x) = \lim_{x\to-\infty} (x^5 - 9x^3)$$

Using the properties of limits, we can rewrite the expression as:

$$\lim_{x\to-\infty} f(x) = \lim_{x\to-\infty} x^5(1 - 9x^{-2})$$

Since $x^5$ approaches negative infinity as $x$ approaches negative infinity, we can conclude that:

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

This means that as $x$ approaches negative infinity, the value of $f(x)$ approaches negative infinity.

Behavior of the Function as $x$ Approaches Positive Infinity

As $x$ approaches positive infinity, the value of $f(x)$ can be determined by analyzing the behavior of the function. Since the function is a polynomial function, we can use the properties of polynomial functions to determine the behavior of the function as $x$ approaches positive infinity.

The function $f(x) = x^5 - 9x^3$ is an odd-degree polynomial function, which means that it will have a horizontal asymptote at $y = 0$. This means that as $x$ approaches positive infinity, the value of $f(x)$ will approach positive infinity.

To determine the exact behavior of the function as $x$ approaches positive infinity, we can use the following inequality:

$$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} (x^5 - 9x^3)$$

Using the properties of limits, we can rewrite the expression as:

$$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} x^5(1 - 9x^{-2})$$

Since $x^5$ approaches positive infinity as $x$ approaches positive infinity, we can conclude that:

$$\lim_{x\to\infty} f(x) = \infty$$

This means that as $x$ approaches positive infinity, the value of $f(x)$ approaches positive infinity.

Key Features of the Graph

The graph of the function $f(x) = x^5 - 9x^3$ has several key features that can be determined by analyzing the behavior of the function as $x$ approaches negative and positive infinity.

• Horizontal Asymptote: The graph of the function has a horizontal asymptote at $y = 0$. This means that as $x$ approaches negative and positive infinity, the value of $f(x)$ approaches $0$.
• End Behavior: The graph of the function has end behavior that is consistent with the behavior of the function as $x$ approaches negative and positive infinity. As $x$ approaches negative infinity, the value of $f(x)$ approaches negative infinity, and as $x$ approaches positive infinity, the value of $f(x)$ approaches positive infinity.
• Critical Points: The graph of the function has critical points at $x = 0$ and $x = \pm \sqrt[3]{9}$. These critical points are where the function changes from increasing to decreasing or vice versa.

Conclusion

In this article, we analyzed the key features of the graph of the function $f(x) = x^5 - 9x^3$. We determined the behavior of the function as $x$ approaches negative and positive infinity, and identified the key features of the graph. The graph of the function has a horizontal asymptote at $y = 0$, end behavior that is consistent with the behavior of the function as $x$ approaches negative and positive infinity, and critical points at $x = 0$ and $x = \pm \sqrt[3]{9}$.

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Q: What is the horizontal asymptote of the graph of $f(x) = x^5 - 9x^3$?

A: The horizontal asymptote of the graph of $f(x) = x^5 - 9x^3$ is $y = 0$. This means that as $x$ approaches negative and positive infinity, the value of $f(x)$ approaches $0$.

Q: What is the end behavior of the graph of $f(x) = x^5 - 9x^3$?

A: The end behavior of the graph of $f(x) = x^5 - 9x^3$ is that as $x$ approaches negative infinity, the value of $f(x)$ approaches negative infinity, and as $x$ approaches positive infinity, the value of $f(x)$ approaches positive infinity.

Q: What are the critical points of the graph of $f(x) = x^5 - 9x^3$?

A: The critical points of the graph of $f(x) = x^5 - 9x^3$ are $x = 0$ and $x = \pm \sqrt[3]{9}$. These critical points are where the function changes from increasing to decreasing or vice versa.

Q: Is the graph of $f(x) = x^5 - 9x^3$ continuous?

A: Yes, the graph of $f(x) = x^5 - 9x^3$ is continuous. This means that the function has no gaps or jumps in its graph.

Q: Is the graph of $f(x) = x^5 - 9x^3$ differentiable?

A: Yes, the graph of $f(x) = x^5 - 9x^3$ is differentiable. This means that the function has a derivative at every point in its domain.

Q: What is the domain of the function $f(x) = x^5 - 9x^3$?

A: The domain of the function $f(x) = x^5 - 9x^3$ is all real numbers. This means that the function is defined for every value of $x$.

Q: What is the range of the function $f(x) = x^5 - 9x^3$?

A: The range of the function $f(x) = x^5 - 9x^3$ is all real numbers. This means that the function can take on any value.

Q: Is the function $f(x) = x^5 - 9x^3$ an even function or an odd function?

A: The function $f(x) = x^5 - 9x^3$ is an odd function. This means that $f(-x) = -f(x)$ for all values of $x$.

Q: Is the function $f(x) = x^5 - 9x^3$ a polynomial function?

A: Yes, the function $f(x) = x^5 - 9x^3$ is a polynomial function. This means that it can be written in the form $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where $a_n \neq 0$.

Q: What is the degree of the polynomial function $f(x) = x^5 - 9x^3$?

A: The degree of the polynomial function $f(x) = x^5 - 9x^3$ is $5$. This means that the highest power of $x$ in the function is $5$.

Q: Is the function $f(x) = x^5 - 9x^3$ a rational function?

A: No, the function $f(x) = x^5 - 9x^3$ is not a rational function. This means that it cannot be written in the form $p(x)/q(x)$, where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$.

Q: Is the function $f(x) = x^5 - 9x^3$ an exponential function?

A: No, the function $f(x) = x^5 - 9x^3$ is not an exponential function. This means that it cannot be written in the form $a^x$ or $a^x + b$, where $a$ and $b$ are constants and $a \neq 0$.

Q: Is the function $f(x) = x^5 - 9x^3$ a trigonometric function?

A: No, the function $f(x) = x^5 - 9x^3$ is not a trigonometric function. This means that it cannot be written in the form $\sin x$, $\cos x$, $\tan x$, or any other trigonometric function.

Q: Is the function $f(x) = x^5 - 9x^3$ a logarithmic function?

A: No, the function $f(x) = x^5 - 9x^3$ is not a logarithmic function. This means that it cannot be written in the form $\log x$ or $\log_b x$, where $b$ is a positive constant and $b \neq 1$.

Q: Is the function $f(x) = x^5 - 9x^3$ a power function?

A: Yes, the function $f(x) = x^5 - 9x^3$ is a power function. This means that it can be written in the form $x^n$, where $n$ is a constant.

Q: What is the period of the function $f(x) = x^5 - 9x^3$?

A: The period of the function $f(x) = x^5 - 9x^3$ is undefined. This means that the function does not have a periodic behavior.

Q: Is the function $f(x) = x^5 - 9x^3$ an even function or an odd function?

A: The function $f(x) = x^5 - 9x^3$ is an odd function. This means that $f(-x) = -f(x)$ for all values of $x$.

Q: Is the function $f(x) = x^5 - 9x^3$ a polynomial function?

A: Yes, the function $f(x) = x^5 - 9x^3$ is a polynomial function. This means that it can be written in the form $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where $a_n \neq 0$.

Q: What is the degree of the polynomial function $f(x) = x^5 - 9x^3$?

A: The degree of the polynomial function $f(x) = x^5 - 9x^3$ is $5$. This means that the highest power of $x$ in the function is $5$.

Q: Is the function $f(x) = x^5 - 9x^3$ a rational function?

A: No, the function $f(x) = x^5 - 9x^3$ is not a rational function. This means that it cannot be written in the form $p(x)/q(x)$, where $p(x)$ and $q(x)$ are polynomials and $q(x) \neq 0$.

Q: Is the function $f(x) = x^5 - 9x^3$ an exponential function?

A: No, the function $f(x) = x^5 - 9x^3$ is not an exponential function. This means that it cannot be written in the form $a^x$ or $a^x + b$, where $a$ and $b$ are constants and $a \neq 0$.

Q: Is the function $f(x) = x^5 - 9x^3$ a trigonometric function?

A: No, the function $f(x) = x^5 - 9x^3$ is not a trigonometric function. This means that it cannot be written in the form $\sin x$, $\cos x$, $\tan x$, or any other trigonometric function.

Q: Is the function $f(x) = x^5 - 9x^3$ a logarithmic function?

A: No, the function $f(x) = x^5 - 9x^3$ is not a logarithmic function. This means that it cannot be written in the form $\log x$ or $\log_b x$, where $b$ is a positive constant and $b \neq 1$.

Q: Is the function $f(x) = x^5 - 9x^3$ a power function?

A: Yes, the function $f(x) = x^5 - 9x^3$ is a power function. This means that it can be written in the form $x^n$, where $n$ is a constant.