Identify The Domain And Range Of The Function $y = 4 \sqrt{x}$.A. Domain: $\{x \mid X \in \mathbb{R}\}$ Range: \$\{y \mid Y \in \mathbb{R}\}$[/tex\]B. Domain: $\{x \mid X \in \mathbb{R}, X \geq 0\}$ Range:

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Introduction

In mathematics, functions are used to describe the relationship between two variables. The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In this article, we will focus on identifying the domain and range of the function $y = 4 \sqrt{x}$.

What is the Domain of a Function?

The domain of a function is the set of all possible input values. In other words, it is the set of all values of x for which the function is defined. To determine the domain of a function, we need to consider the restrictions on the input values.

Restrictions on Input Values

There are several types of restrictions on input values:

  • Non-negative values: If the function involves the square root of x, then x must be non-negative, i.e., x ≥ 0.
  • Real numbers: If the function involves real numbers, then x must be a real number.
  • Integer values: If the function involves integer values, then x must be an integer.

Domain of the Function $y = 4 \sqrt{x}$

The function $y = 4 \sqrt{x}$ involves the square root of x. Therefore, x must be non-negative, i.e., x ≥ 0. Additionally, x must be a real number. Therefore, the domain of the function is:

{x∣x∈R,x≥0}\{x \mid x \in \mathbb{R}, x \geq 0\}

What is the Range of a Function?

The range of a function is the set of all possible output values. In other words, it is the set of all values of y for which the function is defined.

Determining the Range of a Function

To determine the range of a function, we need to consider the following:

  • Minimum and maximum values: If the function has a minimum or maximum value, then the range is the set of all values between the minimum and maximum values.
  • Increasing or decreasing values: If the function is increasing or decreasing, then the range is the set of all values that the function can take.

Range of the Function $y = 4 \sqrt{x}$

The function $y = 4 \sqrt{x}$ is an increasing function. Therefore, the range of the function is the set of all values that the function can take. Since the square root of x is always non-negative, the range of the function is:

{y∣y∈R,y≥0}\{y \mid y \in \mathbb{R}, y \geq 0\}

Conclusion

In conclusion, the domain of the function $y = 4 \sqrt{x}$ is the set of all non-negative real numbers, i.e., ${x \mid x \in \mathbb{R}, x \geq 0}$. The range of the function is the set of all non-negative real numbers, i.e., ${y \mid y \in \mathbb{R}, y \geq 0}$. Understanding the domain and range of a function is essential in mathematics and is used in various applications, including calculus, algebra, and statistics.

Example Problems

  1. Find the domain and range of the function $y = 3 \sqrt{x}$.
  2. Find the domain and range of the function $y = 2 \sqrt{x + 1}$.
  3. Find the domain and range of the function $y = \sqrt{x - 2}$.

Solutions

  1. The domain of the function $y = 3 \sqrt{x}$ is ${x \mid x \in \mathbb{R}, x \geq 0}$. The range of the function is ${y \mid y \in \mathbb{R}, y \geq 0}$.
  2. The domain of the function $y = 2 \sqrt{x + 1}$ is ${x \mid x \in \mathbb{R}, x \geq -1}$. The range of the function is ${y \mid y \in \mathbb{R}, y \geq 0}$.
  3. The domain of the function $y = \sqrt{x - 2}$ is ${x \mid x \in \mathbb{R}, x \geq 2}$. The range of the function is ${y \mid y \in \mathbb{R}, y \geq 0}$.

Final Thoughts

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values. In other words, it is the set of all values of x for which the function is defined.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values. In other words, it is the set of all values of y for which the function is defined.

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

A: To determine the domain of a function, you need to consider the restrictions on the input values. These restrictions may include:

  • Non-negative values: If the function involves the square root of x, then x must be non-negative, i.e., x ≥ 0.
  • Real numbers: If the function involves real numbers, then x must be a real number.
  • Integer values: If the function involves integer values, then x must be an integer.

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

A: To determine the range of a function, you need to consider the following:

  • Minimum and maximum values: If the function has a minimum or maximum value, then the range is the set of all values between the minimum and maximum values.
  • Increasing or decreasing values: If the function is increasing or decreasing, then the range is the set of all values that the function can take.

Q: What is the domain and range of the function $y = 4 \sqrt{x}$?

A: The domain of the function $y = 4 \sqrt{x}$ is ${x \mid x \in \mathbb{R}, x \geq 0}$. The range of the function is ${y \mid y \in \mathbb{R}, y \geq 0}$.

Q: What is the domain and range of the function $y = 3 \sqrt{x}$?

A: The domain of the function $y = 3 \sqrt{x}$ is ${x \mid x \in \mathbb{R}, x \geq 0}$. The range of the function is ${y \mid y \in \mathbb{R}, y \geq 0}$.

Q: What is the domain and range of the function $y = 2 \sqrt{x + 1}$?

A: The domain of the function $y = 2 \sqrt{x + 1}$ is ${x \mid x \in \mathbb{R}, x \geq -1}$. The range of the function is ${y \mid y \in \mathbb{R}, y \geq 0}$.

Q: What is the domain and range of the function $y = \sqrt{x - 2}$?

A: The domain of the function $y = \sqrt{x - 2}$ is ${x \mid x \in \mathbb{R}, x \geq 2}$. The range of the function is ${y \mid y \in \mathbb{R}, y \geq 0}$.

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

A: To graph a function with a domain and range, you need to consider the following:

  • Domain: The domain of the function is the set of all possible input values. You can graph the domain by plotting the x-values on the x-axis.
  • Range: The range of the function is the set of all possible output values. You can graph the range by plotting the y-values on the y-axis.

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

A: Understanding the domain and range of a function is essential in mathematics and is used in various applications, including calculus, algebra, and statistics. It helps you to:

  • Determine the behavior of a function: By understanding the domain and range of a function, you can determine its behavior and make predictions about its output values.
  • Solve equations and inequalities: Understanding the domain and range of a function helps you to solve equations and inequalities involving the function.
  • Graph functions: Understanding the domain and range of a function helps you to graph the function and visualize its behavior.

Conclusion

In conclusion, understanding the domain and range of a function is essential in mathematics. By considering the restrictions on input values and the behavior of the function, you can determine the domain and range of a function. This knowledge is used in various applications, including calculus, algebra, and statistics.