Applying the Properties of the Square Root Function to the Given Function

Oct 13, 2025 by pynkocean.com 4 views

Introduction

In mathematics, functions are used to describe the relationship between variables. The domain and range of a function are crucial concepts that help us understand the behavior of the function. In this article, we will focus on the domain and range of the function $f(x) = \sqrt{x} + 2$ . We will explore the different options and determine the correct domain and range of the function.

What is the Domain of a Function?

The domain of a function is the set of all possible input values (x-values) that the function can accept without resulting in an undefined or imaginary output. In other words, it is the set of all possible values of x for which the function is defined.

What is the Range of a Function?

The range of a function is the set of all possible output values (y-values) that the function can produce for the given input values. It is the set of all possible values of y for which the function is defined.

Analyzing the Function $f(x) = \sqrt{x} + 2$

The given function is $f(x) = \sqrt{x} + 2$ . To determine the domain and range of this function, we need to consider the properties of the square root function.

The Square Root Function

The square root function, denoted by $\sqrt{x}$ , is defined only for non-negative real numbers. This means that the input value x must be greater than or equal to 0 for the square root function to be defined.

Applying the Properties of the Square Root Function to the Given Function

Now, let's apply the properties of the square root function to the given function $f(x) = \sqrt{x} + 2$ . Since the square root function is defined only for non-negative real numbers, the input value x must be greater than or equal to 0 for the function to be defined.

Determining the Domain of the Function

Based on the properties of the square root function, we can conclude that the domain of the function $f(x) = \sqrt{x} + 2$ is the set of all non-negative real numbers, which can be represented as $[0, \infty)$ .

Determining the Range of the Function

To determine the range of the function, we need to consider the output values of the function for the given input values. Since the square root function is defined only for non-negative real numbers, the output value of the function will also be non-negative.

The Constant Term +2

However, we need to consider the constant term +2 in the function. This term will shift the output values of the function upwards by 2 units.

The Correct Range of the Function

Based on the above analysis, we can conclude that the range of the function $f(x) = \sqrt{x} + 2$ is the set of all non-negative real numbers shifted upwards by 2 units, which can be represented as $[2, \infty)$ .

Conclusion

In conclusion, the domain of the function $f(x) = \sqrt{x} + 2$ is the set of all non-negative real numbers, represented as $[0, \infty)$ , and the range of the function is the set of all non-negative real numbers shifted upwards by 2 units, represented as $[2, \infty)$ .

Answer

The correct answer is:

A. - Domain: $[0, \infty)$

Range: $[2, \infty)$
Frequently Asked Questions (FAQs) about the Domain and Range of a Function ====================================================================================

Introduction

In our previous article, we discussed the domain and range of the function $f(x) = \sqrt{x} + 2$ . We explored the different options and determined the correct domain and range of the function. In this article, we will answer some frequently asked questions (FAQs) about the domain and range of a function.

Q: What is the difference between the domain and range of a function?

A: The domain of a function is the set of all possible input values (x-values) that the function can accept without resulting in an undefined or imaginary output. The range of a function is the set of all possible output values (y-values) that the function can produce for the given input values.

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

A: To determine the domain of a function, you need to consider the properties of the function and the values of x for which the function is defined. For example, if the function involves a square root, you need to ensure that the input value x is non-negative.

Q: What is the domain of the function $f(x) = \sqrt{x} + 2$ ?

A: The domain of the function $f(x) = \sqrt{x} + 2$ is the set of all non-negative real numbers, represented as $[0, \infty)$ .

Q: What is the range of the function $f(x) = \sqrt{x} + 2$ ?

A: The range of the function $f(x) = \sqrt{x} + 2$ is the set of all non-negative real numbers shifted upwards by 2 units, represented as $[2, \infty)$ .

Q: Can the domain and range of a function be the same?

A: Yes, the domain and range of a function can be the same. For example, if the function is a linear function, the domain and range may be the same.

Q: How do I determine the range of a function?

A: To determine the range of a function, you need to consider the output values of the function for the given input values. You can use the properties of the function and the values of y to determine the range.

Q: What is the importance of the domain and range of a function?

A: The domain and range of a function are crucial concepts in mathematics. They help us understand the behavior of the function and determine the possible input and output values.

Q: Can the domain and range of a function be negative?

A: No, the domain and range of a function cannot be negative. The domain and range are always non-negative, as they represent the possible input and output values of the function.

Q: How do I graph the domain and range of a function?

A: To graph the domain and range of a function, you can use a coordinate plane. The domain is represented on the x-axis, and the range is represented on the y-axis.

Conclusion

In conclusion, the domain and range of a function are crucial concepts in mathematics. They help us understand the behavior of the function and determine the possible input and output values. We hope that this article has answered some of your frequently asked questions about the domain and range of a function.