Choose The Function That Has Domain { X \neq -3 $}$ And Range { Y \neq 2 $}$.A. { F(x) = \frac{x+2}{x+3} $}$B. { F(x) = \frac{2x+1}{x+3} $}$C. { F(x) = \frac{x-3}{x+2} $}$

Mar 11, 2025 by ADMIN 172 views

**Domain and Range of Functions: Understanding the Basics**

Introduction

In mathematics, functions are used to describe the relationship between two variables. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. Understanding the domain and range of a function is crucial in mathematics, as it helps us to determine the validity of a function and its behavior. In this article, we will explore the concept of domain and range, and how to determine them for a given function.

What is Domain and Range?

The domain of a function is the set of all possible input values, or x-values, that the function can accept. It is the set of all possible values of x that make the function defined. On the other hand, the range of a function is the set of all possible output values, or y-values, that the function can produce. It is the set of all possible values of y that the function can take.

Determining Domain and Range

To determine the domain and range of a function, we need to consider the following:

Vertical Asymptotes: A vertical asymptote is a vertical line that the function approaches but never touches. It is a value of x that makes the function undefined. To determine the domain, we need to find the values of x that make the function undefined.
Horizontal Asymptotes: A horizontal asymptote is a horizontal line that the function approaches as x approaches infinity or negative infinity. It is a value of y that the function approaches as x approaches infinity or negative infinity. To determine the range, we need to find the values of y that the function approaches as x approaches infinity or negative infinity.
Holes: A hole is a point where the function is undefined, but the function approaches that point as x approaches a certain value. To determine the domain, we need to find the values of x that make the function undefined, but the function approaches that value as x approaches a certain value.

Solving the Problem

Now, let's solve the problem given in the discussion category. We are given three functions, and we need to determine which function has a domain of ${$ x \neq -3 $}$ and a range of ${$ y \neq 2 $}$.

Function A

${$ f(x) = \frac{x+2}{x+3} $}$

To determine the domain of this function, we need to find the values of x that make the function undefined. The function is undefined when the denominator is equal to zero, i.e., when ${$ x+3 = 0 $}$. Solving for x, we get ${$ x = -3 $}$. Therefore, the domain of this function is ${$ x \neq -3 $}$.

To determine the range of this function, we need to find the values of y that the function approaches as x approaches infinity or negative infinity. As x approaches infinity, the function approaches ${$ \frac{1}{1} = 1 $}$. As x approaches negative infinity, the function approaches ${$ \frac{1}{1} = 1 $}$. Therefore, the range of this function is ${$ y \neq 1 $}$.

Function B

${$ f(x) = \frac{2x+1}{x+3} $}$

To determine the range of this function, we need to find the values of y that the function approaches as x approaches infinity or negative infinity. As x approaches infinity, the function approaches ${$ \frac{2}{1} = 2 $}$. As x approaches negative infinity, the function approaches ${$ \frac{2}{1} = 2 $}$. Therefore, the range of this function is ${$ y \neq 2 $}$.

Function C

${$ f(x) = \frac{x-3}{x+2} $}$

To determine the domain of this function, we need to find the values of x that make the function undefined. The function is undefined when the denominator is equal to zero, i.e., when ${$ x+2 = 0 $}$. Solving for x, we get ${$ x = -2 $}$. Therefore, the domain of this function is ${$ x \neq -2 $}$.

Conclusion

In conclusion, the function that has a domain of ${$ x \neq -3 $}$ and a range of ${$ y \neq 2 $}$ is Function B. This is because the domain of Function B is ${$ x \neq -3 $}$, and the range of Function B is ${$ y \neq 2 $}$.

References

[1] "Domain and Range of Functions" by Khan Academy
[2] "Domain and Range" by Math Open Reference
[3] "Functions" by Wolfram MathWorld

Additional Resources

[1] "Domain and Range of Functions" by IXL
[2] "Domain and Range" by Purplemath
[3] "Functions" by Mathway
Domain and Range of Functions: Q&A =====================================

Introduction

In our previous article, we discussed the concept of domain and range of functions, and how to determine them for a given function. In this article, we will answer some frequently asked questions related to domain and range of functions.

Q: What is the difference between domain and range of a function?

A: The domain of a function is the set of all possible input values, or x-values, that the function can accept. The range of a function is the set of all possible output values, or y-values, that the function can produce.

Q: How do I determine the domain of a function?

A: To determine the domain of a function, you need to find the values of x that make the function undefined. This can be done by finding the values of x that make the denominator of the function equal to zero.

Q: How do I determine the range of a function?

A: To determine the range of a function, you need to find the values of y that the function approaches as x approaches infinity or negative infinity.

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that the function approaches but never touches. It is a value of x that makes the function undefined.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a horizontal line that the function approaches as x approaches infinity or negative infinity. It is a value of y that the function approaches as x approaches infinity or negative infinity.

Q: What is a hole in a function?

A: A hole in a function is a point where the function is undefined, but the function approaches that point as x approaches a certain value.

Q: How do I determine the domain and range of a rational function?

A: To determine the domain and range of a rational function, you need to find the values of x that make the denominator equal to zero, and the values of y that the function approaches as x approaches infinity or negative infinity.

Q: How do I determine the domain and range of a polynomial function?

A: To determine the domain and range of a polynomial function, you need to find the values of x that make the function undefined, and the values of y that the function approaches as x approaches infinity or negative infinity.

Q: What is the difference between a domain and a range of a function?

Q: Can a function have a domain of all real numbers?

A: Yes, a function can have a domain of all real numbers. This means that the function can accept any real number as input.

Q: Can a function have a range of all real numbers?

A: Yes, a function can have a range of all real numbers. This means that the function can produce any real number as output.

Q: What is the relationship between the domain and range of a function?

A: The domain and range of a function are related in that the domain of a function determines the possible input values, and the range of a function determines the possible output values.

Conclusion

In conclusion, the domain and range of a function are important concepts in mathematics that help us to understand the behavior of a function. By understanding the domain and range of a function, we can determine the possible input and output values of the function.

References

[1] "Domain and Range of Functions" by Khan Academy
[2] "Domain and Range" by Math Open Reference
[3] "Functions" by Wolfram MathWorld

Additional Resources

[1] "Domain and Range of Functions" by IXL
[2] "Domain and Range" by Purplemath
[3] "Functions" by Mathway