Which Of The Following Is True About The Function Below?$\frac{1}{\sqrt{x+4}}$A. Its Domain Is $(-4, \infty$\] And Its Range Is $(-\infty, \infty$\].B. Its Domain Is $(-4, \infty$\] And Its Range Is $(0,

Introduction

In mathematics, functions are used to describe the relationship between variables. Understanding the domain and range of a function is crucial in various mathematical and real-world applications. In this article, we will analyze the given function $\frac{1}{\sqrt{x+4}}$ and determine its domain and range.

Domain and Range Analysis

The given function is $\frac{1}{\sqrt{x+4}}$. To determine its domain and range, we need to consider the restrictions on the input and output values.

Domain Analysis

The domain of a function is the set of all possible input values for which the function is defined. In the case of the given function, we need to consider the restrictions on the input value $x$.

• Square Root Restriction: The expression inside the square root must be non-negative, i.e., $x+4 \geq 0$. This implies that $x \geq -4$.
• Denominator Restriction: The denominator of the fraction cannot be zero, i.e., $\sqrt{x+4} \neq 0$. This implies that $x+4 > 0$, which further implies that $x > -4$.

Combining these restrictions, we can conclude that the domain of the function is $(-4, \infty)$.

Range Analysis

The range of a function is the set of all possible output values for which the function is defined. In the case of the given function, we need to consider the restrictions on the output value.

• Square Root Restriction: The expression inside the square root must be non-negative, i.e., $x+4 \geq 0$. This implies that $x \geq -4$.
• Fraction Restriction: The fraction must be non-negative, i.e., $\frac{1}{\sqrt{x+4}} \geq 0$. This implies that $\sqrt{x+4} > 0$, which further implies that $x > -4$.

Combining these restrictions, we can conclude that the range of the function is $(0, \infty)$.

Conclusion

In conclusion, the domain of the function $\frac{1}{\sqrt{x+4}}$ is $(-4, \infty)$, and its range is $(0, \infty)$. Therefore, the correct answer is:

• Option B: Its domain is $(-4, \infty)$ and its range is $(0, \infty)$.

Final Answer

Introduction

In the previous article, we analyzed the function $\frac{1}{\sqrt{x+4}}$ and determined its domain and range. In this article, we will provide a Q&A section to further clarify the concepts and provide additional insights.

Q&A

Q1: What is the domain of the function $\frac{1}{\sqrt{x+4}}$?

A1: The domain of the function $\frac{1}{\sqrt{x+4}}$ is $(-4, \infty)$.

Q2: Why is the domain of the function $\frac{1}{\sqrt{x+4}}$ restricted to $(-4, \infty)$?

A2: The domain of the function $\frac{1}{\sqrt{x+4}}$ is restricted to $(-4, \infty)$ because the expression inside the square root must be non-negative, i.e., $x+4 \geq 0$. This implies that $x \geq -4$. Additionally, the denominator of the fraction cannot be zero, i.e., $\sqrt{x+4} \neq 0$. This implies that $x+4 > 0$, which further implies that $x > -4$.

Q3: What is the range of the function $\frac{1}{\sqrt{x+4}}$?

A3: The range of the function $\frac{1}{\sqrt{x+4}}$ is $(0, \infty)$.

Q4: Why is the range of the function $\frac{1}{\sqrt{x+4}}$ restricted to $(0, \infty)$?

A4: The range of the function $\frac{1}{\sqrt{x+4}}$ is restricted to $(0, \infty)$ because the fraction must be non-negative, i.e., $\frac{1}{\sqrt{x+4}} \geq 0$. This implies that $\sqrt{x+4} > 0$, which further implies that $x > -4$.

Q5: Can the function $\frac{1}{\sqrt{x+4}}$ be defined for $x = -4$?

A5: No, the function $\frac{1}{\sqrt{x+4}}$ cannot be defined for $x = -4$ because the expression inside the square root would be zero, i.e., $x+4 = 0$. This would make the denominator of the fraction zero, which is not allowed.

Q6: Can the function $\frac{1}{\sqrt{x+4}}$ be defined for $x < -4$?

A6: No, the function $\frac{1}{\sqrt{x+4}}$ cannot be defined for $x < -4$ because the expression inside the square root would be negative, i.e., $x+4 < 0$. This would make the square root undefined.

Conclusion

In conclusion, the function $\frac{1}{\sqrt{x+4}}$ has a domain of $(-4, \infty)$ and a range of $(0, \infty)$. The domain is restricted to $(-4, \infty)$ because the expression inside the square root must be non-negative, and the denominator of the fraction cannot be zero. The range is restricted to $(0, \infty)$ because the fraction must be non-negative.

Final Answer

The final answer is that the domain of the function $\frac{1}{\sqrt{x+4}}$ is $(-4, \infty)$, and its range is $(0, \infty)$.