Select The Correct Answer From Each Drop-down Menu.Consider The End Behavior Of This Function, And Then Complete The Statements.$h(x)=-\frac{1}{5}|x-3|+4$As $x$ Approaches Negative Infinity, $h(x$\] Approaches

Introduction

When analyzing a function, it's essential to consider its end behavior, which refers to the behavior of the function as the input values approach positive or negative infinity. In this article, we will explore the end behavior of the given function $h(x)=-\frac{1}{5}|x-3|+4$ and complete the statements.

The Function $h(x)=-\frac{1}{5}|x-3|+4$

The given function is a piecewise function, which means it has different expressions for different intervals of the input variable $x$. The absolute value function $|x-3|$ is defined as:

$$|x-3| = \begin{cases} x-3, & \text{if } x \geq 3 \\ -(x-3), & \text{if } x < 3 \end{cases}$$

Substituting this definition into the original function, we get:

$$h(x) = \begin{cases} -\frac{1}{5}(x-3)+4, & \text{if } x \geq 3 \\ -\frac{1}{5}(-(x-3))+4, & \text{if } x < 3 \end{cases}$$

Simplifying the expressions, we get:

$$h(x) = \begin{cases} -\frac{1}{5}x+\frac{13}{5}, & \text{if } x \geq 3 \\ \frac{1}{5}x+\frac{13}{5}, & \text{if } x < 3 \end{cases}$$

End Behavior as $x$ Approaches Negative Infinity

To analyze the end behavior of the function as $x$ approaches negative infinity, we need to consider the behavior of the function for large negative values of $x$. Since the function is defined as $\frac{1}{5}x+\frac{13}{5}$ for $x < 3$, we will focus on this expression.

As $x$ approaches negative infinity, the term $\frac{1}{5}x$ dominates the expression, and the constant term $\frac{13}{5}$ becomes negligible. Therefore, as $x$ approaches negative infinity, the function approaches:

$$\lim_{x\to-\infty} h(x) = \lim_{x\to-\infty} \left(\frac{1}{5}x+\frac{13}{5}\right) = -\infty$$

End Behavior as $x$ Approaches Positive Infinity

To analyze the end behavior of the function as $x$ approaches positive infinity, we need to consider the behavior of the function for large positive values of $x$. Since the function is defined as $-\frac{1}{5}x+\frac{13}{5}$ for $x \geq 3$, we will focus on this expression.

As $x$ approaches positive infinity, the term $-\frac{1}{5}x$ dominates the expression, and the constant term $\frac{13}{5}$ becomes negligible. Therefore, as $x$ approaches positive infinity, the function approaches:

$$\lim_{x\to\infty} h(x) = \lim_{x\to\infty} \left(-\frac{1}{5}x+\frac{13}{5}\right) = -\infty$$

Conclusion

In conclusion, as $x$ approaches negative infinity, the function $h(x)=-\frac{1}{5}|x-3|+4$ approaches negative infinity. Similarly, as $x$ approaches positive infinity, the function also approaches negative infinity.

Select the Correct Answer from Each Drop-Down Menu

Based on the analysis, the correct answer is:

• As $x$ approaches negative infinity, $h(x)$ approaches negative infinity.
• As $x$ approaches positive infinity, $h(x)$ approaches negative infinity.

Introduction

In our previous article, we explored the end behavior of the function $h(x)=-\frac{1}{5}|x-3|+4$. We analyzed the behavior of the function as $x$ approaches negative and positive infinity. In this article, we will answer some frequently asked questions related to the end behavior of a function.

Q: What is the end behavior of a function?

A: The end behavior of a function refers to the behavior of the function as the input values approach positive or negative infinity.

Q: How do I determine the end behavior of a function?

A: To determine the end behavior of a function, you need to analyze the function's expression and identify the term that dominates the expression as the input values approach positive or negative infinity.

Q: What is the difference between the end behavior and the limit of a function?

A: The end behavior of a function refers to the behavior of the function as the input values approach positive or negative infinity, while the limit of a function refers to the value that the function approaches as the input values approach a specific value.

Q: Can the end behavior of a function be different from the limit of the function?

A: Yes, the end behavior of a function can be different from the limit of the function. For example, a function may approach a specific value as the input values approach a certain value, but its end behavior may be different.

Q: How do I determine the end behavior of a piecewise function?

A: To determine the end behavior of a piecewise function, you need to analyze each expression separately and identify the term that dominates the expression as the input values approach positive or negative infinity.

Q: Can the end behavior of a function be affected by the presence of absolute value or other non-linear terms?

A: Yes, the end behavior of a function can be affected by the presence of absolute value or other non-linear terms. These terms can change the behavior of the function as the input values approach positive or negative infinity.

Q: How do I use the end behavior of a function to make predictions about its behavior?

A: You can use the end behavior of a function to make predictions about its behavior by analyzing the function's expression and identifying the term that dominates the expression as the input values approach positive or negative infinity.

Q: Can the end behavior of a function be used to determine the function's asymptotes?

A: Yes, the end behavior of a function can be used to determine the function's asymptotes. The end behavior of a function can help you identify the horizontal or vertical asymptotes of the function.

Conclusion

In conclusion, the end behavior of a function is an essential concept in mathematics that helps us understand the behavior of a function as the input values approach positive or negative infinity. By analyzing the function's expression and identifying the term that dominates the expression, we can determine the end behavior of a function and make predictions about its behavior.

Select the Correct Answer from Each Drop-Down Menu

Based on the Q&A, the correct answers are:

• The end behavior of a function refers to the behavior of the function as the input values approach positive or negative infinity.
• The end behavior of a function can be different from the limit of the function.
• The end behavior of a piecewise function can be determined by analyzing each expression separately and identifying the term that dominates the expression as the input values approach positive or negative infinity.
• The end behavior of a function can be affected by the presence of absolute value or other non-linear terms.
• The end behavior of a function can be used to make predictions about its behavior by analyzing the function's expression and identifying the term that dominates the expression as the input values approach positive or negative infinity.
• The end behavior of a function can be used to determine the function's asymptotes.