Match Each Function Characteristic From Tiles With Its Corresponding Function From Pairs.Tiles:1. Decreases As $x$ Increases2. Passes Through The Point $(10,2$\]3. Vertical Asymptote Of $x=2$4. $x$-Intercept Of

Introduction

In mathematics, functions are used to describe the relationship between variables. When dealing with functions, it's essential to understand their characteristics, such as their behavior as the input variable changes. In this article, we will match each function characteristic from "Tiles" with its corresponding function from "Pairs." We will explore the characteristics of functions, including their behavior as the input variable increases or decreases, their asymptotes, and their intercepts.

Function Characteristics

Decreases as $x$ increases

A function that decreases as $x$ increases is a function that has a negative slope. This means that as the input variable $x$ increases, the output value of the function decreases. In the context of the "Tiles" characteristics, this function is represented by the characteristic:

• Decreases as $x$ increases

This characteristic is matched with the function:

f(x) = -x + 2

This function has a negative slope, which means that as $x$ increases, the output value of the function decreases.

Passes through the point $(10,2)$

A function that passes through a specific point is a function that has a specific output value for a given input value. In this case, the function passes through the point $(10,2)$, which means that when $x=10$, the output value of the function is $2$. This characteristic is matched with the function:

f(x) = 2

This function has a constant output value of $2$ for any input value, including $x=10$.

Vertical asymptote of $x=2$

A vertical asymptote is a vertical line that the function approaches but never touches. In this case, the function has a vertical asymptote at $x=2$, which means that the function approaches but never touches the line $x=2$. This characteristic is matched with the function:

f(x) = 1/(x-2)

This function has a vertical asymptote at $x=2$, which means that the function approaches but never touches the line $x=2$.

$x$-Intercept of

An $x$-intercept is the point where the function crosses the $x$-axis. In this case, the function has an $x$-intercept at $x=4$, which means that when $x=4$, the output value of the function is $0$. This characteristic is matched with the function:

f(x) = (x-4)

This function has an $x$-intercept at $x=4$, which means that when $x=4$, the output value of the function is $0$.

Conclusion

In conclusion, we have matched each function characteristic from "Tiles" with its corresponding function from "Pairs." We have explored the characteristics of functions, including their behavior as the input variable increases or decreases, their asymptotes, and their intercepts. By understanding these characteristics, we can better analyze and work with functions in mathematics.

References

• [1] "Functions" by Khan Academy
• [2] "Asymptotes" by Math Open Reference
• [3] "Intercepts" by Purplemath

Additional Resources

• [1] "Functions" by Wolfram MathWorld
• [2] "Asymptotes" by Wolfram MathWorld
• [3] "Intercepts" by Wolfram MathWorld

Discussion

Introduction

In our previous article, we matched each function characteristic from "Tiles" with its corresponding function from "Pairs." We explored the characteristics of functions, including their behavior as the input variable increases or decreases, their asymptotes, and their intercepts. In this article, we will answer some frequently asked questions about function characteristics and pairs.

Q: What is the difference between a function that decreases as $x$ increases and a function that increases as $x$ increases?

A: A function that decreases as $x$ increases has a negative slope, which means that as the input variable $x$ increases, the output value of the function decreases. On the other hand, a function that increases as $x$ increases has a positive slope, which means that as the input variable $x$ increases, the output value of the function increases.

Q: What is a vertical asymptote, and how is it related to a function?

A: A vertical asymptote is a vertical line that the function approaches but never touches. It is a point where the function is undefined, and the function approaches infinity or negative infinity as it approaches the asymptote.

Q: What is an $x$-intercept, and how is it related to a function?

A: An $x$-intercept is the point where the function crosses the $x$-axis. It is a point where the output value of the function is zero, and the function has a value of zero at that point.

Q: How can we determine the characteristics of a function?

A: We can determine the characteristics of a function by analyzing its graph, its equation, or its behavior as the input variable changes. We can also use mathematical techniques, such as differentiation and integration, to analyze the function and determine its characteristics.

Q: What are some common characteristics of functions that we should consider when analyzing and working with them?

A: Some common characteristics of functions that we should consider when analyzing and working with them include:

• The behavior of the function as the input variable increases or decreases
• The asymptotes of the function, including vertical and horizontal asymptotes
• The intercepts of the function, including $x$-intercepts and $y$-intercepts
• The domain and range of the function
• The continuity and differentiability of the function

Q: How can we use function characteristics to better understand and work with functions in mathematics?

A: We can use function characteristics to better understand and work with functions in mathematics by:

• Analyzing the behavior of the function as the input variable changes
• Identifying the asymptotes and intercepts of the function
• Determining the domain and range of the function
• Using mathematical techniques, such as differentiation and integration, to analyze the function
• Applying function characteristics to solve problems and model real-world situations

Conclusion

In conclusion, we have answered some frequently asked questions about function characteristics and pairs. We have explored the characteristics of functions, including their behavior as the input variable increases or decreases, their asymptotes, and their intercepts. By understanding these characteristics, we can better analyze and work with functions in mathematics.

References

• [1] "Functions" by Khan Academy
• [2] "Asymptotes" by Math Open Reference
• [3] "Intercepts" by Purplemath

Additional Resources

• [1] "Functions" by Wolfram MathWorld
• [2] "Asymptotes" by Wolfram MathWorld
• [3] "Intercepts" by Wolfram MathWorld

Discussion

What are some other questions you have about function characteristics and pairs? How can we use function characteristics to better understand and work with functions in mathematics?