Which Function Has A Domain Where { X \neq 3 $}$ And A Range Where { Y \neq 2 $}$?A. { F(x) = \frac{x-5}{x+3} $}$B. { F(x) = \frac{2(x+5)}{x+3} $}$C. { F(x) = \frac{2(x+5)}{x-3} $} D . \[ D. \[ D . \[ F(x)

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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 that the function can produce. When dealing with functions, it's essential to understand the domain and range, as they can significantly impact the behavior and properties of the function.

Analyzing the Given Functions

We are given four functions, and we need to determine which one has a domain where { x \neq 3 $}$ and a range where { y \neq 2 $}$. To do this, we'll analyze each function individually and examine their domains and ranges.

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, we are given that the domain is { x \neq 3 $}$, which means that { x=-3 $}$ is not the only value that needs to be excluded.

To determine the range of function A, we need to consider the possible output values. Since the numerator is { x-5 $}$ and the denominator is { x+3 $}$, the function can produce any real number except for { y=2 $}$. This is because when { x=-3 $}$, the function is undefined, and when { x=3 $}$, the function produces { y=-2 $}$.

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

The domain of function B is similar to that of function A, as the denominator is also { x+3 $}$. However, the numerator is { 2(x+5) $}$, which means that the function can produce any real number except for { y=2 $}$. This is because when { x=-3 $}$, the function is undefined, and when { x=3 $}$, the function produces { y=4 $}$.

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

The domain of function C is different from that of functions A and B, as the denominator is { x-3 $}$. This means that the function is defined for all real numbers except for { x=3 $}$. However, the range of function C is similar to that of functions A and B, as the function can produce any real number except for { y=2 $}$.

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

The domain of function D is similar to that of functions A and B, as the denominator is also { x+3 $}$. However, the numerator is { 2(x+5) $}$, which means that the function can produce any real number except for { y=2 $}$. This is because when { x=-3 $}$, the function is undefined, and when { x=3 $}$, the function produces { y=4 $}$.

Conclusion

Based on our analysis, we can conclude that function B has a domain where { x \neq 3 $}$ and a range where { y \neq 2 $}$. This is because the function is defined for all real numbers except for { x=3 $}$, and it can produce any real number except for { y=2 $}$.

Key Takeaways

  • The domain of a function is the set of all possible input values for which the function is defined.
  • The range of a function is the set of all possible output values that the function can produce.
  • When dealing with functions, it's essential to understand the domain and range, as they can significantly impact the behavior and properties of the function.
  • The domain and range of a function can be determined by analyzing the function's numerator and denominator.

Final Answer

The final answer is function B: { f(x) = \frac{2(x+5)}{x+3} $}$.
Q&A: Understanding the Domain and Range of a Function

Introduction

In our previous article, we discussed the concept of domain and range in functions. We analyzed four different functions and determined which one has a domain where { x \neq 3 $}$ and a range where { y \neq 2 $}$. In this article, we'll answer some frequently asked questions related to 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's the set of all possible x-values that can be plugged into the function.

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

A: To determine the domain of a function, you need to examine the function's numerator and denominator. If the denominator is equal to zero, you need to exclude that value from the domain. Additionally, if the function has any restrictions on the input values, you need to exclude those values as well.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values that the function can produce. In other words, it's the set of all possible y-values that can be produced by the function.

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

A: To determine the range of a function, you need to examine the function's numerator and denominator. If the function has any restrictions on the output values, you need to exclude those values from the range.

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 because the domain and range of a function are determined by the function's numerator and denominator, and there's no inherent restriction that prevents a function from having a domain where { x \neq 3 $}$ and a range where { y \neq 2 $}$.

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

A: To determine if a function has a domain where { x \neq 3 $}$ and a range where { y \neq 2 $}$, you need to examine the function's numerator and denominator. If the denominator is equal to zero when { x=3 $}$, and the function produces a value other than { y=2 $}$ when { x=3 $}$, then the function has a domain where { x \neq 3 $}$ and a range where { y \neq 2 $}$.

Q: Can a function have a domain where { x \neq 3 $}$ and a range where { y \neq 2 $}$ if the function is undefined at { x=3 $}$?

A: Yes, a function can have a domain where { x \neq 3 $}$ and a range where { y \neq 2 $}$ even if the function is undefined at { x=3 $}$. This is because the domain and range of a function are determined by the function's numerator and denominator, and the fact that the function is undefined at { x=3 $}$ does not necessarily mean that the function has a domain where { x \neq 3 $}$ and a range where { y \neq 2 $}$.

Conclusion

In conclusion, the domain and range of a function are determined by the function's numerator and denominator. A function can have a domain where { x \neq 3 $}$ and a range where { y \neq 2 $}$ if the denominator is equal to zero when { x=3 $}$ and the function produces a value other than { y=2 $}$ when { x=3 $}$. We hope this article has helped you understand the concept of domain and range in functions.

Key Takeaways

  • The domain of a function is the set of all possible input values for which the function is defined.
  • The range of a function is the set of all possible output values that the function can produce.
  • A function can have a domain where { x \neq 3 $}$ and a range where { y \neq 2 $}$ if the denominator is equal to zero when { x=3 $}$ and the function produces a value other than { y=2 $}$ when { x=3 $}$.

Final Answer

The final answer is that a function can have a domain where { x \neq 3 $}$ and a range where { y \neq 2 $}$ if the denominator is equal to zero when { x=3 $}$ and the function produces a value other than { y=2 $}$ when { x=3 $}$.