The Domain Of $y=\sqrt{x-5}-1$ Is:A. All Real Numbers.B. X ≥ − 5 X \geq -5 X ≥ − 5 C. X ≥ − 1 X \geq -1 X ≥ − 1 D. X ≥ 1 X \geq 1 X ≥ 1 E. X ≥ 5 X \geq 5 X ≥ 5

Mar 9, 2025 by ADMIN 162 views

**The Domain of a Square Root Function: Understanding the Restrictions**

Introduction

When dealing with square root functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values for which the function is defined. In the case of a square root function, the domain is restricted to values that make the expression inside the square root non-negative. In this article, we will explore the domain of the function $y=\sqrt{x-5}-1$ and examine the restrictions that apply to its input values.

Understanding the Square Root Function

The square root function is defined as $\sqrt{x} = y$, where $x$ is the input value and $y$ is the output value. The square root function is only defined for non-negative values of $x$ , i.e., $x \geq 0$ . This is because the square root of a negative number is undefined in the real number system.

Restrictions on the Input Value

In the given function $y=\sqrt{x-5}-1$, the expression inside the square root is $x-5$ . For the function to be defined, the expression inside the square root must be non-negative, i.e., $x-5 \geq 0$ . This implies that $x \geq 5$ .

Analyzing the Options

Let's analyze the options provided:

A. All real numbers: This option is incorrect because the function is not defined for all real numbers. The expression inside the square root must be non-negative.

B. $x \geq -5$ : This option is incorrect because the function is not defined for values of $x$ less than 5.

C. $x \geq -1$ : This option is incorrect because the function is not defined for values of $x$ less than 5.

D. $x \geq 1$ : This option is incorrect because the function is not defined for values of $x$ less than 5.

E. $x \geq 5$ : This option is correct because the expression inside the square root must be non-negative, i.e., $x-5 \geq 0$ , which implies that $x \geq 5$ .

Conclusion

In conclusion, the domain of the function $y=\sqrt{x-5}-1$ is restricted to values of $x$ that make the expression inside the square root non-negative. This implies that $x \geq 5$ . Therefore, the correct answer is option E.

Final Thoughts

Understanding the domain of a function is crucial in mathematics, as it helps us determine the input values for which the function is defined. In this article, we explored the domain of the function $y=\sqrt{x-5}-1$ and examined the restrictions that apply to its input values. By analyzing the options provided, we determined that the correct answer is option E, $x \geq 5$ . This knowledge can be applied to various mathematical problems and is essential for understanding the behavior of functions.

Common Mistakes to Avoid

When dealing with square root functions, it's essential to remember that the expression inside the square root must be non-negative. This is a common mistake that many students make. To avoid this mistake, always check the expression inside the square root and ensure that it is non-negative.

Real-World Applications

Understanding the domain of a function has real-world applications in various fields, such as engineering, economics, and computer science. For example, in engineering, understanding the domain of a function can help us determine the input values for which a system is stable. In economics, understanding the domain of a function can help us determine the input values for which a model is valid. In computer science, understanding the domain of a function can help us determine the input values for which a program is correct.

Conclusion

In conclusion, understanding the domain of a function is crucial in mathematics and has real-world applications in various fields. By analyzing the options provided, we determined that the correct answer is option E, $x \geq 5$ . This knowledge can be applied to various mathematical problems and is essential for understanding the behavior of functions.

Introduction

In our previous article, we explored the domain of the function $y=\sqrt{x-5}-1$ and examined the restrictions that apply to its input values. In this article, we will answer some frequently asked questions related to the domain of a square root function.

Q: What is the domain of a square root function?

A: The domain of a square root function is the set of all possible input values for which the function is defined. In the case of a square root function, the domain is restricted to values that make the expression inside the square root non-negative.

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

A: To determine the domain of a square root function, you need to check the expression inside the square root and ensure that it is non-negative. If the expression is non-negative, then the function is defined for all values of $x$ that make the expression non-negative.

Q: What happens if the expression inside the square root is negative?

A: If the expression inside the square root is negative, then the function is undefined for that value of $x$ . This is because the square root of a negative number is undefined in the real number system.

Q: Can I have a negative value inside the square root?

A: No, you cannot have a negative value inside the square root. The square root of a negative number is undefined in the real number system.

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 for which the function is defined. The range of a function is the set of all possible output values of the function. In other words, the domain is the set of all possible $x$ values, while the range is the set of all possible $y$ values.

Q: Can I have a square root function with a negative value inside the square root and still have a defined function?

A: No, you cannot have a square root function with a negative value inside the square root and still have a defined function. The square root of a negative number is undefined in the real number system.

Q: How do I know if a function is defined or undefined?

A: To determine if a function is defined or undefined, you need to check the expression inside the square root and ensure that it is non-negative. If the expression is non-negative, then the function is defined. If the expression is negative, then the function is undefined.

Q: Can I have a square root function with a zero value inside the square root?

A: Yes, you can have a square root function with a zero value inside the square root. In this case, the function is defined for all values of $x$ that make the expression non-negative.

Q: What is the significance of the domain of a function?

A: The domain of a function is crucial in mathematics, as it helps us determine the input values for which the function is defined. Understanding the domain of a function is essential for understanding the behavior of functions and making predictions about the output values of the function.

Conclusion

In conclusion, understanding the domain of a square root function is crucial in mathematics. By answering these frequently asked questions, we hope to have provided a better understanding of the domain of a square root function and its significance in mathematics.

Common Mistakes to Avoid

Real-World Applications

Conclusion

In conclusion, understanding the domain of a square root function is crucial in mathematics and has real-world applications in various fields. By answering these frequently asked questions, we hope to have provided a better understanding of the domain of a square root function and its significance in mathematics.