Which Function Has The Domain $x \geq -11$?A. Y = X + 11 + 5 Y = \sqrt{x + 11} + 5 Y = X + 11 ​ + 5 B. Y = X − 11 + 5 Y = \sqrt{x - 11} + 5 Y = X − 11 ​ + 5 C. Y = X + 5 − 11 Y = \sqrt{x + 5} - 11 Y = X + 5 ​ − 11 D. Y = X + 5 + 11 Y = \sqrt{x + 5} + 11 Y = X + 5 ​ + 11

Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. When dealing with square root functions, it's essential to consider the domain carefully, as the expression inside the square root must be non-negative. In this article, we will analyze the given functions and determine which one has the domain $x \geq -11$.

Understanding Square Root Functions

A square root function is defined as $y = \sqrt{x}$, where $x$ is the input value and $y$ is the output value. The expression inside the square root must be non-negative, i.e., $x \geq 0$. If the input value is negative, the function is undefined.

Analyzing the Given Functions

We are given four square root functions:

A. $y = \sqrt{x + 11} + 5$ B. $y = \sqrt{x - 11} + 5$ C. $y = \sqrt{x + 5} - 11$ D. $y = \sqrt{x + 5} + 11$

To determine which function has the domain $x \geq -11$, we need to analyze each function individually.

Function A: $y = \sqrt{x + 11} + 5$

The expression inside the square root is $x + 11$. For this expression to be non-negative, we must have $x + 11 \geq 0$. Solving for $x$, we get $x \geq -11$. Therefore, the domain of function A is $x \geq -11$.

Function B: $y = \sqrt{x - 11} + 5$

The expression inside the square root is $x - 11$. For this expression to be non-negative, we must have $x - 11 \geq 0$. Solving for $x$, we get $x \geq 11$. Therefore, the domain of function B is $x \geq 11$, which is not equal to $x \geq -11$.

Function C: $y = \sqrt{x + 5} - 11$

The expression inside the square root is $x + 5$. For this expression to be non-negative, we must have $x + 5 \geq 0$. Solving for $x$, we get $x \geq -5$. Therefore, the domain of function C is $x \geq -5$, which is not equal to $x \geq -11$.

Function D: $y = \sqrt{x + 5} + 11$

The expression inside the square root is $x + 5$. For this expression to be non-negative, we must have $x + 5 \geq 0$. Solving for $x$, we get $x \geq -5$. Therefore, the domain of function D is $x \geq -5$, which is not equal to $x \geq -11$.

Conclusion

Based on our analysis, we can conclude that the only function that has the domain $x \geq -11$ is function A: $y = \sqrt{x + 11} + 5$.

Final Answer

Introduction

In our previous article, we analyzed the given square root functions and determined which one has the domain $x \geq -11$. In this article, we will provide a Q&A section to further clarify the concepts and provide additional insights.

Q&A

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 expression inside the square root must be 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 analyze the expression inside the square root. If the expression is non-negative, then the domain is all real numbers greater than or equal to the minimum value of the expression.

Q: What is the minimum value of the expression inside the square root?

A: The minimum value of the expression inside the square root is the value that makes the expression equal to zero. For example, in the function $y = \sqrt{x + 11} + 5$, the minimum value of the expression inside the square root is $-11$.

Q: How do I know if a square root function is defined for a given input value?

A: To determine if a square root function is defined for a given input value, you need to check if the expression inside the square root is non-negative. If it is, then the function is defined for that input value.

Q: Can a square root function have a domain that is a single value?

A: Yes, a square root function can have a domain that is a single value. For example, the function $y = \sqrt{x}$ has a domain of $x \geq 0$, but it is only defined for $x = 0$.

Q: Can a square root function have a domain that is a range of values?

A: Yes, a square root function can have a domain that is a range of values. For example, the function $y = \sqrt{x + 11} + 5$ has a domain of $x \geq -11$, which is a range of values.

Q: How do I graph a square root function?

A: To graph a square root function, you need to plot the points that satisfy the function and connect them with a smooth curve. You can also use a graphing calculator or software to graph the function.

Q: Can a square root function be used to model real-world phenomena?

A: Yes, a square root function can be used to model real-world phenomena. For example, the growth of a population can be modeled using a square root function.

Conclusion

In this article, we provided a Q&A section to further clarify the concepts and provide additional insights on the domain analysis of square root functions. We hope that this article has been helpful in understanding the domain of square root functions.

Final Answer

The final answer is A.