Describe The End Behavior Of The Function $g(x)=\frac{1}{x^2}$.A. As $x$ Goes To Negative Infinity, $g(x$\] Goes To Positive Infinity. As $x$ Goes To Positive Infinity, $g(x$\] Goes To Positive Infinity.B. As

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Introduction

In mathematics, the end behavior of a function refers to the behavior of the function as the input values approach positive or negative infinity. Understanding the end behavior of a function is crucial in various mathematical and real-world applications, such as graphing, optimization, and modeling. In this article, we will discuss the end behavior of the function $g(x)=\frac{1}{x^2}$.

The Function $g(x)=\frac{1}{x^2}$

The function $g(x)=\frac{1}{x^2}$ is a rational function, where the numerator is a constant (1) and the denominator is a quadratic expression ($x^2$). To understand the end behavior of this function, we need to analyze its behavior as $x$ approaches positive or negative infinity.

End Behavior as $x$ Approaches Positive Infinity

As $x$ approaches positive infinity, the denominator $x^2$ becomes increasingly large. Since the numerator is a constant (1), the value of the function $g(x)$ will decrease as the denominator increases. However, the function will never actually reach zero, because the denominator will always be positive and will continue to increase without bound.

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

$$\lim_{x\to\infty}g(x)=\lim_{x\to\infty}\frac{1}{x^2}$$

Using the properties of limits, we can rewrite this limit as:

$$\lim_{x\to\infty}g(x)=\lim_{x\to\infty}\frac{1}{x^2}=\lim_{x\to\infty}\frac{1}{x^2}\cdot\frac{1}{x^2}=\lim_{x\to\infty}\frac{1}{x^4}$$

Since the denominator $x^4$ approaches infinity as $x$ approaches infinity, the value of the function $g(x)$ will approach zero.

End Behavior as $x$ Approaches Negative Infinity

As $x$ approaches negative infinity, the denominator $x^2$ becomes increasingly large, just like in the case where $x$ approaches positive infinity. However, the function $g(x)$ will behave differently as $x$ approaches negative infinity.

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

$$\lim_{x\to-\infty}g(x)=\lim_{x\to-\infty}\frac{1}{x^2}$$

Using the properties of limits, we can rewrite this limit as:

$$\lim_{x\to-\infty}g(x)=\lim_{x\to-\infty}\frac{1}{x^2}=\lim_{x\to-\infty}\frac{1}{x^2}\cdot\frac{1}{x^2}=\lim_{x\to-\infty}\frac{1}{x^4}$$

Since the denominator $x^4$ approaches infinity as $x$ approaches negative infinity, the value of the function $g(x)$ will approach zero.

Conclusion

In conclusion, the end behavior of the function $g(x)=\frac{1}{x^2}$ is as follows:

• As $x$ approaches positive infinity, the function $g(x)$ approaches zero.
• As $x$ approaches negative infinity, the function $g(x)$ approaches zero.

The function $g(x)$ has a horizontal asymptote at $y=0$, which means that as $x$ approaches positive or negative infinity, the function $g(x)$ will approach zero.

Graph of the Function

The graph of the function $g(x)=\frac{1}{x^2}$ is a hyperbola that opens along the x-axis. The graph has a horizontal asymptote at $y=0$, which means that as $x$ approaches positive or negative infinity, the graph will approach the x-axis.

Real-World Applications

The function $g(x)=\frac{1}{x^2}$ has various real-world applications, such as:

• Modeling the behavior of physical systems, such as the motion of a particle or the vibration of a spring.
• Analyzing the behavior of electrical circuits, such as the behavior of a resistor or a capacitor.
• Understanding the behavior of economic systems, such as the behavior of supply and demand.

Limitations

The function $g(x)=\frac{1}{x^2}$ has some limitations, such as:

• The function is not defined at $x=0$, because the denominator is zero.
• The function is not defined for negative values of $x$, because the denominator is negative.

Conclusion

In conclusion, the end behavior of the function $g(x)=\frac{1}{x^2}$ is as follows:

• As $x$ approaches positive infinity, the function $g(x)$ approaches zero.
• As $x$ approaches negative infinity, the function $g(x)$ approaches zero.

The function $g(x)$ has a horizontal asymptote at $y=0$, which means that as $x$ approaches positive or negative infinity, the function $g(x)$ will approach zero.

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Introduction

In our previous article, we discussed the end behavior of the function $g(x)=\frac{1}{x^2}$. In this article, we will answer some frequently asked questions about the end behavior of this function.

Q: What is the end behavior of the function $g(x)=\frac{1}{x^2}$ as $x$ approaches positive infinity?

A: As $x$ approaches positive infinity, the function $g(x)$ approaches zero. This is because the denominator $x^2$ becomes increasingly large, causing the value of the function to decrease.

Q: What is the end behavior of the function $g(x)=\frac{1}{x^2}$ as $x$ approaches negative infinity?

A: As $x$ approaches negative infinity, the function $g(x)$ approaches zero. This is because the denominator $x^2$ becomes increasingly large, causing the value of the function to decrease.

Q: Why does the function $g(x)=\frac{1}{x^2}$ have a horizontal asymptote at $y=0$?

A: The function $g(x)=\frac{1}{x^2}$ has a horizontal asymptote at $y=0$ because as $x$ approaches positive or negative infinity, the value of the function approaches zero. This means that the function will never actually reach zero, but it will get arbitrarily close to it.

Q: Can the function $g(x)=\frac{1}{x^2}$ be defined at $x=0$?

A: No, the function $g(x)=\frac{1}{x^2}$ cannot be defined at $x=0$ because the denominator is zero. This means that the function is not continuous at $x=0$.

Q: Can the function $g(x)=\frac{1}{x^2}$ be defined for negative values of $x$?

A: No, the function $g(x)=\frac{1}{x^2}$ cannot be defined for negative values of $x$ because the denominator is negative. This means that the function is not defined for $x<0$.

Q: What are some real-world applications of the function $g(x)=\frac{1}{x^2}$?

A: Some real-world applications of the function $g(x)=\frac{1}{x^2}$ include:

• Modeling the behavior of physical systems, such as the motion of a particle or the vibration of a spring.
• Analyzing the behavior of electrical circuits, such as the behavior of a resistor or a capacitor.
• Understanding the behavior of economic systems, such as the behavior of supply and demand.

Q: What are some limitations of the function $g(x)=\frac{1}{x^2}$?

A: Some limitations of the function $g(x)=\frac{1}{x^2}$ include:

• The function is not defined at $x=0$.
• The function is not defined for negative values of $x$.
• The function has a horizontal asymptote at $y=0$, which means that it will never actually reach zero.

Conclusion

In conclusion, the end behavior of the function $g(x)=\frac{1}{x^2}$ is as follows:

• As $x$ approaches positive infinity, the function $g(x)$ approaches zero.
• As $x$ approaches negative infinity, the function $g(x)$ approaches zero.

The function $g(x)$ has a horizontal asymptote at $y=0$, which means that as $x$ approaches positive or negative infinity, the function $g(x)$ will approach zero.