Finding the Range of Function C

Oct 13, 2025 by thesweetdesigns.com 120 views

Which Function Has a Domain Where ${$ x \neq 3$}$ and a Range Where ${$ y \neq 2$}$?

Understanding the Domain and Range of a Function

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. On the other hand, the range of a function is the set of all possible output values. In this article, we will explore which function has a domain where ${$ x \neq 3$}$ and a range where ${$ y \neq 2$}$.

Analyzing the Functions

To determine which function has a domain where ${$ x \neq 3$}$ and a range where ${$ y \neq 2$}$, we need to analyze each function individually.

Function A: ${$ f(x) = \frac{x-5}{x+3}$}$

The domain of a function is the set of all possible input values for which the function is defined. In the case of function A, the denominator is ${$ x+3$}$. Since the denominator cannot be equal to zero, we must exclude ${$ x=-3$}$ from the domain. However, the problem states that the domain is ${$ x \neq 3$}$, not ${$ x \neq -3$}$. Therefore, function A does not meet the criteria.

Function B: ${$ f(x) = \frac{2(x+5)}{x+3}$}$

The domain of function B is also the set of all possible input values for which the function is defined. In this case, the denominator is still ${$ x+3$}$. Since the denominator cannot be equal to zero, we must exclude ${$ x=-3$}$ from the domain. However, the problem states that the domain is ${$ x \neq 3$}$, not ${$ x \neq -3$}$. Therefore, function B does not meet the criteria.

Function C: ${$ f(x) = \frac{2(x+5)}{x-3}$}$

The domain of function C is the set of all possible input values for which the function is defined. In this case, the denominator is ${$ x-3$}$. Since the denominator cannot be equal to zero, we must exclude ${$ x=3$}$ from the domain. This matches the problem statement, which states that the domain is ${$ x \neq 3$}$. Therefore, function C meets the criteria.

Function D: ${$ f(x) = \frac{2(x+5)}{x+3}$}$

The domain of function D is the set of all possible input values for which the function is defined. In this case, the denominator is ${$ x+3$}$. Since the denominator cannot be equal to zero, we must exclude ${$ x=-3$}$ from the domain. However, the problem states that the domain is ${$ x \neq 3$}$, not ${$ x \neq -3$}$. Therefore, function D does not meet the criteria.

Conclusion

Based on the analysis of each function, we can conclude that function C, ${$ f(x) = \frac{2(x+5)}{x-3}$}$, has a domain where ${$ x \neq 3$}$ and a range where ${$ y \neq 2$}$.

Understanding the Range of a Function

The range of a function is the set of all possible output values. In the case of function C, we need to determine the range of the function.

Finding the Range of Function C

To find the range of function C, we can start by simplifying the function. We can rewrite function C as ${$ f(x) = \frac{2x+10}{x-3}$}$.

Simplifying the Function

To simplify the function, we can divide both the numerator and the denominator by their greatest common divisor. In this case, the greatest common divisor is 1. Therefore, we cannot simplify the function further.

Finding the Range

To find the range of the function, we can analyze the behavior of the function as x approaches positive infinity and negative infinity.

Behavior as x Approaches Positive Infinity

As x approaches positive infinity, the denominator ${$ x-3$}$ approaches positive infinity. Therefore, the function approaches ${$ \frac{2x+10}{x-3} \approx \frac{2x}{x} = 2$}$ as x approaches positive infinity.

Behavior as x Approaches Negative Infinity

As x approaches negative infinity, the denominator ${$ x-3$}$ approaches negative infinity. Therefore, the function approaches ${$ \frac{2x+10}{x-3} \approx \frac{2x}{x} = 2$}$ as x approaches negative infinity.

Conclusion

Based on the analysis of the behavior of the function as x approaches positive infinity and negative infinity, we can conclude that the range of function C is ${$ y \neq 2$}$.

Conclusion

In conclusion, function C, ${$ f(x) = \frac{2(x+5)}{x-3}$}$, has a domain where ${$ x \neq 3$}$ and a range where ${$ y \neq 2$}$. This function meets the criteria specified in the problem.
Q&A: Understanding the Domain and Range of a Function

Introduction

In our previous article, we explored which function has a domain where ${$ x \neq 3$}$ and a range where ${$ y \neq 2$}$. We analyzed each function individually and concluded that function C, ${$ f(x) = \frac{2(x+5)}{x-3}$}$, meets the criteria. In this article, we will answer some frequently asked questions about the domain and range of a function.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible x-values for which the function is defined.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values. In other words, it is the set of all possible y-values for which the function is defined.

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

A: To determine the domain of a function, you need to identify any values of x that would make the denominator of the function equal to zero. These values must be excluded from the domain.

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

A: To determine the range of a function, you need to analyze the behavior of the function as x approaches positive infinity and negative infinity. This will give you an idea of the possible output values of the function.

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

A: The domain of a function is the set of all possible input values, while the range of a function is the set of all possible output values.

Q: Can a function have a domain where ${$ x \neq 3$}$ and a range where ${$ y \neq 2$}$?

A: Yes, a function can have a domain where ${$ x \neq 3$}$ and a range where ${$ y \neq 2$}$. This is the case for function C, ${$ f(x) = \frac{2(x+5)}{x-3}$}$.

Q: How do I graph a function with a domain where ${$ x \neq 3$}$ and a range where ${$ y \neq 2$}$?

A: To graph a function with a domain where ${$ x \neq 3$}$ and a range where ${$ y \neq 2$}$, you need to exclude the value x = 3 from the graph and also exclude the value y = 2 from the graph.

Q: Can a function have a domain where ${$ x \neq 3$}$ and a range where ${$ y \neq 2$}$ and also be a linear function?

A: Yes, a function can have a domain where ${$ x \neq 3$}$ and a range where ${$ y \neq 2$}$ and also be a linear function. This is the case for function C, ${$ f(x) = \frac{2(x+5)}{x-3}$}$.

Conclusion

In conclusion, understanding the domain and range of a function is crucial in mathematics. By answering these frequently asked questions, we hope to have provided a better understanding of the domain and range of a function. If you have any further questions, please do not hesitate to ask.

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