What Is The Domain Of The Following Function: $f(x) = \frac{\sqrt{x+2}}{x-2}$Please Indicate Your Answer By Selecting The Correct Option Below:- A. All Real Numbers - B. All Real Numbers Except X = 2- C. X > -2 And X ≠ 2- D. X ≥ -2 And X ≠ 2

Introduction

When dealing with functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the set of all possible x-values that the function can accept without resulting in an undefined or imaginary output. In this article, we'll explore the domain of the function $f(x) = \frac{\sqrt{x+2}}{x-2}$ and determine the correct answer from the given options.

What is the Domain of a Function?

The domain of a function is the set of all possible input values (x-values) for which the function is defined. To determine the domain of a function, we need to consider the following:

• Square Root Function: The square root function is defined only for non-negative values. In other words, the expression inside the square root must be greater than or equal to zero.
• Fraction Function: The fraction function is defined only when the denominator is not equal to zero.

Analyzing the Given Function

The given function is $f(x) = \frac{\sqrt{x+2}}{x-2}$. To determine the domain of this function, we need to consider the following:

• Square Root Expression: The expression inside the square root is $x+2$. For this expression to be non-negative, we must have $x+2 \geq 0$. Solving this inequality, we get $x \geq -2$.
• Denominator Expression: The denominator expression is $x-2$. For this expression to be non-zero, we must have $x-2 \neq 0$. Solving this equation, we get $x \neq 2$.

Determining the Domain

Based on the analysis above, we can conclude that the domain of the function $f(x) = \frac{\sqrt{x+2}}{x-2}$ is the set of all real numbers greater than or equal to -2, except for x = 2.

Conclusion

In conclusion, the domain of the function $f(x) = \frac{\sqrt{x+2}}{x-2}$ is the set of all real numbers greater than or equal to -2, except for x = 2. This can be represented mathematically as $x \geq -2$ and $x \neq 2$.

Answer

Based on the analysis above, the correct answer is:

• D. x ≥ -2 and x ≠ 2

This answer indicates that the domain of the function $f(x) = \frac{\sqrt{x+2}}{x-2}$ is the set of all real numbers greater than or equal to -2, except for x = 2.

Final Thoughts

Introduction

In our previous article, we explored the concept of the domain of a function and determined the domain of the function $f(x) = \frac{\sqrt{x+2}}{x-2}$. In this article, we'll answer some frequently asked questions related to the domain of a function.

Q&A

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-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 consider the following:

• Square Root Function: The square root function is defined only for non-negative values. In other words, the expression inside the square root must be greater than or equal to zero.
• Fraction Function: The fraction function is defined only when the denominator is not equal to zero.

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

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range of a function is the set of all possible output values (y-values) that the function can produce.

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

A: Yes, a function can have a domain of all real numbers. For example, the function $f(x) = x^2$ has a domain of all real numbers.

Q: Can a function have a domain of all real numbers except for a certain value?

A: Yes, a function can have a domain of all real numbers except for a certain value. For example, the function $f(x) = \frac{1}{x}$ has a domain of all real numbers except for x = 0.

Q: How do I determine the domain of a function with a square root in the numerator?

A: To determine the domain of a function with a square root in the numerator, you need to consider the following:

• Square Root Expression: The expression inside the square root must be non-negative.
• Denominator Expression: The denominator expression must be non-zero.

Q: How do I determine the domain of a function with a fraction in the numerator?

A: To determine the domain of a function with a fraction in the numerator, you need to consider the following:

• Fraction Expression: The fraction expression must be non-zero.
• Denominator Expression: The denominator expression must be non-zero.

Q: Can a function have a domain of all real numbers except for a certain interval?

A: Yes, a function can have a domain of all real numbers except for a certain interval. For example, the function $f(x) = \frac{1}{x^2-4}$ has a domain of all real numbers except for the interval $(-2, 2)$.

Q: How do I determine the domain of a function with a square root in the denominator?

A: To determine the domain of a function with a square root in the denominator, you need to consider the following:

• Square Root Expression: The expression inside the square root must be non-negative.
• Denominator Expression: The denominator expression must be non-zero.

Q: Can a function have a domain of all real numbers except for a certain value and a certain interval?

A: Yes, a function can have a domain of all real numbers except for a certain value and a certain interval. For example, the function $f(x) = \frac{1}{x^2-4}$ has a domain of all real numbers except for x = 2 and the interval $(-2, 2)$.

Conclusion

In conclusion, the domain of a function is the set of all possible input values (x-values) for which the function is defined. By considering the square root and fraction functions, we can determine the domain of a function. We've answered some frequently asked questions related to the domain of a function, and we hope this article has been helpful in understanding the concept of the domain of a function.

Final Thoughts

Understanding the domain of a function is crucial in mathematics, as it helps us determine the set of all possible input values for which the function is defined. By analyzing the given function and considering the square root and fraction functions, we can determine the domain of the function $f(x) = \frac{\sqrt{x+2}}{x-2}$ as the set of all real numbers greater than or equal to -2, except for x = 2.