Domain of a Radical Function

Oct 13, 2025 by magicalceylontours.com 63 views

Understanding the Radical Function

A radical function is a type of function that involves a square root or other root. In the given function $$y = \sqrt{x+7} + 5$$, the square root is the primary operation. The function takes an input value $x$ , adds $7$ to it, and then takes the square root of the result. Finally, it adds $5$ to the result.

Domain of a Radical Function

The domain of a function is the set of all possible input values for which the function is defined. In the case of a radical function, the domain is restricted by the fact that the expression inside the square root must be non-negative.

Identifying the Domain of the Given Function

To identify the domain of the given function $$y = \sqrt{x+7} + 5$$, we need to find the values of $x$ for which the expression inside the square root is non-negative.

\sqrt{x+7} \geq 0

Since the square root of a number is non-negative if and only if the number itself is non-negative, we can rewrite the inequality as:

x+7 \geq 0

Subtracting $7$ from both sides of the inequality, we get:

x \geq -7

Therefore, the domain of the given function is all real numbers greater than or equal to $-7$ .

Graphing the Radical Function

To graph the radical function $$y = \sqrt{x+7} + 5$$, we can start by plotting the graph of the function $y = \sqrt{x}$ , which is a square root function with a domain of all non-negative real numbers.

Next, we can shift the graph of $y = \sqrt{x}$ to the left by $7$ units to obtain the graph of $y = \sqrt{x+7}$ .

Finally, we can shift the graph of $y = \sqrt{x+7}$ up by $5$ units to obtain the graph of the given function $$y = \sqrt{x+7} + 5$$.

Graph of the Radical Function

The graph of the radical function $$y = \sqrt{x+7} + 5$$ is a continuous curve that opens upwards. The graph has a minimum point at $x = -7$ , where the function takes on the value $5$ . As $x$ increases, the function increases without bound.

Conclusion

In conclusion, the domain of the given function $$y = \sqrt{x+7} + 5$$ is all real numbers greater than or equal to $-7$ . The graph of the function is a continuous curve that opens upwards, with a minimum point at $x = -7$ .

Answer

The correct answer is:

C. $$x \geq -7$$

Discussion

The domain of a radical function is restricted by the fact that the expression inside the square root must be non-negative. In the case of the given function $$y = \sqrt{x+7} + 5$$, the domain is all real numbers greater than or equal to $-7$ .

References

[1] "Graphing Radical Functions" by Math Open Reference
[2] "Domain of a Radical Function" by Purplemath
[3] "Graphing Functions" by Khan Academy
Q&A: Graphing the Radical Function and Identifying Its Domain ===========================================================

Frequently Asked Questions

Q: What is the domain of a radical function?

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

Q: How do I identify the domain of a radical function?

A: To identify the domain of a radical function, you need to find the values of $x$ for which the expression inside the square root is non-negative. This can be done by setting the expression inside the square root greater than or equal to zero and solving for $x$ .

Q: What is the graph of a radical function?

A: The graph of a radical function is a continuous curve that opens upwards. The graph has a minimum point at the value of $x$ that makes the expression inside the square root equal to zero.

Q: How do I graph a radical function?

A: To graph a radical function, you can start by plotting the graph of the function $y = \sqrt{x}$ , which is a square root function with a domain of all non-negative real numbers. Then, you can shift the graph of $y = \sqrt{x}$ to the left or right by the value of the constant inside the square root, and up or down by the value of the constant outside the square root.

Q: What is the difference between a radical function and a square root function?

A: A radical function is a type of function that involves a square root or other root, while a square root function is a specific type of radical function that involves only the square root. The domain of a radical function is restricted by the fact that the expression inside the square root must be non-negative, while the domain of a square root function is all non-negative real numbers.

Q: Can a radical function have a negative value?

A: No, a radical function cannot have a negative value. The expression inside the square root must be non-negative, so the function can only take on non-negative values.

Q: Can a radical function have a fractional value?

A: Yes, a radical function can have a fractional value. The expression inside the square root can be a fraction, and the function can take on fractional values.

Q: Can a radical function have a complex value?

A: Yes, a radical function can have a complex value. The expression inside the square root can be a complex number, and the function can take on complex values.

Q: How do I find the inverse of a radical function?

A: To find the inverse of a radical function, you need to swap the roles of $x$ and $y$ and solve for $y$ . This can be done by using algebraic manipulations to isolate $y$ .

Q: What is the derivative of a radical function?

A: The derivative of a radical function is a type of function that involves the derivative of the expression inside the square root. The derivative of a radical function can be found using the chain rule and the power rule.

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

A: Yes, a radical function can be used to model real-world phenomena. Radical functions can be used to model situations where the relationship between two variables is not linear, but rather involves a square root or other root.

Q: What are some common applications of radical functions?

A: Some common applications of radical functions include:

Modeling the growth of populations
Modeling the spread of diseases
Modeling the behavior of electrical circuits
Modeling the motion of objects under the influence of gravity

Conclusion

In conclusion, radical functions are a type of function that involves a square root or other root. The domain of a radical function is restricted by the fact that the expression inside the square root must be non-negative. Radical functions can be used to model real-world phenomena and have many applications in fields such as physics, engineering, and economics.

References

[1] "Graphing Radical Functions" by Math Open Reference
[2] "Domain of a Radical Function" by Purplemath
[3] "Graphing Functions" by Khan Academy