What Is The Range Of Y = X + 7 + 5 Y=\sqrt{x+7}+5 Y = X + 7 ​ + 5 ?A. Y ≥ − 5 Y \geq -5 Y ≥ − 5 B. Y ≥ 5 Y \geq 5 Y ≥ 5 C. Y ≥ − 7 Y \geq -7 Y ≥ − 7 D. All Real Numbers

**What is the Range of $y=\sqrt{x+7}+5$?**

Understanding the Problem

The given problem is to find the range of the function $y=\sqrt{x+7}+5$. To solve this problem, we need to understand the concept of the range of a function. The range of a function is the set of all possible output values it can produce for the given input values.

The Function $y=\sqrt{x+7}+5$

The given function is $y=\sqrt{x+7}+5$. This function involves a square root and a constant term. To find the range, we need to consider the domain of the function and the behavior of the square root function.

Domain of the Function

The domain of the function $y=\sqrt{x+7}+5$ is all real numbers $x$ such that $x+7 \geq 0$. This is because the square root of a negative number is not defined in the real number system. Therefore, the domain of the function is $x \geq -7$.

Behavior of the Square Root Function

The square root function $\sqrt{x}$ is an increasing function, meaning that as the input $x$ increases, the output $\sqrt{x}$ also increases. However, the square root function is not defined for negative values of $x$. Therefore, the function $y=\sqrt{x+7}+5$ is also an increasing function for $x \geq -7$.

Finding the Range

To find the range of the function $y=\sqrt{x+7}+5$, we need to consider the minimum and maximum values of the function. Since the function is increasing for $x \geq -7$, the minimum value of the function occurs when $x = -7$. Substituting $x = -7$ into the function, we get:

$$y = \sqrt{-7+7}+5 = \sqrt{0}+5 = 5$$

Therefore, the minimum value of the function is $y = 5$. Since the function is increasing for $x \geq -7$, the maximum value of the function is unbounded. Therefore, the range of the function is all real numbers greater than or equal to $5$.

Conclusion

In conclusion, the range of the function $y=\sqrt{x+7}+5$ is all real numbers greater than or equal to $5$. This is because the function is increasing for $x \geq -7$ and the minimum value of the function is $y = 5$.

Q&A

Q: What is the range of the function $y=\sqrt{x+7}+5$?

A: The range of the function $y=\sqrt{x+7}+5$ is all real numbers greater than or equal to $5$.

Q: Why is the domain of the function $x \geq -7$?

A: The domain of the function $x \geq -7$ because the square root of a negative number is not defined in the real number system.

Q: Is the function $y=\sqrt{x+7}+5$ an increasing function?

A: Yes, the function $y=\sqrt{x+7}+5$ is an increasing function for $x \geq -7$.

Q: What is the minimum value of the function $y=\sqrt{x+7}+5$?

A: The minimum value of the function $y=\sqrt{x+7}+5$ is $y = 5$.

Q: Is the range of the function $y=\sqrt{x+7}+5$ bounded?

A: No, the range of the function $y=\sqrt{x+7}+5$ is unbounded.

Q: What is the final answer to the problem?

A: The final answer to the problem is that the range of the function $y=\sqrt{x+7}+5$ is all real numbers greater than or equal to $5$.