What Is The Domain Of The Function Y = 2 X − 5 Y=2 \sqrt{x-5} Y = 2 X − 5 ​ ?A. X ≥ − 5 X \geq -5 X ≥ − 5 B. X ≥ 2 X \geq 2 X ≥ 2 C. X ≥ 5 X \geq 5 X ≥ 5 D. X ≥ 10 X \geq 10 X ≥ 10

Introduction

When dealing with functions, it's essential to understand the concept of the domain. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In other words, it's the range of values that x can take on. In this article, we'll focus on finding the domain of the function $y=2 \sqrt{x-5}$.

What is the Domain of a Function?

The domain of a function is a critical concept in mathematics, particularly in algebra and calculus. It's essential to understand that the domain of a function is not the same as its range. The range of a function is the set of all possible output values (y-values) for which the function is defined.

Why is the Domain Important?

The domain of a function is crucial because it determines the values of x for which the function is defined. If the domain of a function is not properly defined, it can lead to incorrect results or even undefined values. In the case of the function $y=2 \sqrt{x-5}$, we need to find the values of x for which the expression under the square root is non-negative.

Finding the Domain of the Function $y=2 \sqrt{x-5}$

To find the domain of the function $y=2 \sqrt{x-5}$, we need to consider the expression under the square root, which is $x-5$. For the function to be defined, the expression under the square root must be non-negative.

Step 1: Set Up the Inequality

We start by setting up the inequality $x-5 \geq 0$. This inequality represents the condition for the expression under the square root to be non-negative.

Step 2: Solve the Inequality

To solve the inequality $x-5 \geq 0$, we add 5 to both sides of the inequality, which gives us $x \geq 5$.

Step 3: Consider the Domain of the Square Root Function

The square root function is defined only for non-negative values. Therefore, we need to ensure that the expression under the square root, $x-5$, is non-negative.

Step 4: Combine the Results

Combining the results from the previous steps, we find that the domain of the function $y=2 \sqrt{x-5}$ is $x \geq 5$.

Conclusion

In conclusion, the domain of the function $y=2 \sqrt{x-5}$ is $x \geq 5$. This means that the function is defined for all values of x greater than or equal to 5. The domain of a function is a critical concept in mathematics, and understanding it is essential for working with functions.

Answer

The correct answer is:

• C. $x \geq 5$

Additional Tips and Examples

• When dealing with functions, it's essential to understand the concept of the domain.
• The domain of a function is the set of all possible input values (x-values) for which the function is defined.
• The domain of a function is not the same as its range.
• The range of a function is the set of all possible output values (y-values) for which the function is defined.
• To find the domain of a function, consider the expression under the square root and ensure that it is non-negative.
• The domain of a function can be found by setting up and solving an inequality.

Common Mistakes to Avoid

• Not considering the domain of a function when working with it.
• Assuming that the domain of a function is the same as its range.
• Not ensuring that the expression under the square root is non-negative.
• Not combining the results from previous steps when finding the domain of a function.

Real-World Applications

• Understanding the domain of a function is essential in various fields, including physics, engineering, and economics.
• The domain of a function can be used to model real-world situations, such as population growth, financial transactions, and physical systems.
• The domain of a function can be used to make predictions and decisions based on data.

Final Thoughts

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Q: Why is the domain of a function important?

A: The domain of a function is important because it determines the values of x for which the function is defined. If the domain of a function is not properly defined, it can lead to incorrect results or even undefined values.

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

A: To find the domain of a function, consider the expression under the square root and ensure that it is non-negative. Set up and solve an inequality to find the values of x for which the function is defined.

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) for which the function is defined, while the range of a function is the set of all possible output values (y-values) for which the function is defined.

Q: Can the domain of a function be a single value?

A: Yes, the domain of a function can be a single value. For example, the function $f(x) = \sqrt{x-5}$ has a domain of $x \geq 5$, but the function $f(x) = \sqrt{x-5}$ is only defined for $x = 5$.

Q: Can the domain of a function be an interval?

A: Yes, the domain of a function can be an interval. For example, the function $f(x) = \sqrt{x-5}$ has a domain of $x \geq 5$, which is an interval.

Q: Can the domain of a function be a union of intervals?

A: Yes, the domain of a function can be a union of intervals. For example, the function $f(x) = \sqrt{x-5}$ has a domain of $x \geq 5$, but the function $f(x) = \sqrt{x-5}$ is also defined for $x \leq 5$, so the domain is $x \geq 5 \cup x \leq 5$.

Q: Can the domain of a function be a single point?

A: Yes, the domain of a function can be a single point. For example, the function $f(x) = \sqrt{x-5}$ has a domain of $x = 5$.

Q: Can the domain of a function be a union of single points?

A: Yes, the domain of a function can be a union of single points. For example, the function $f(x) = \sqrt{x-5}$ has a domain of $x = 5 \cup x = 10$.

Q: Can the domain of a function be a union of intervals and single points?

A: Yes, the domain of a function can be a union of intervals and single points. For example, the function $f(x) = \sqrt{x-5}$ has a domain of $x \geq 5 \cup x = 10$.

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

A: To determine the domain of a function with a square root, consider the expression under the square root and ensure that it is non-negative. Set up and solve an inequality to find the values of x for which the function is defined.

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

A: To determine the domain of a function with a cube root, consider the expression under the cube root and ensure that it is non-negative. Set up and solve an inequality to find the values of x for which the function is defined.

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

A: To determine the domain of a function with a rational expression, consider the values of x that make the denominator equal to zero. Set up and solve an inequality to find the values of x for which the function is defined.

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

A: To determine the domain of a function with a trigonometric expression, consider the values of x that make the expression undefined. Set up and solve an inequality to find the values of x for which the function is defined.

Q: Can the domain of a function be a complex number?

A: Yes, the domain of a function can be a complex number. For example, the function $f(z) = \sqrt{z-5}$ has a domain of $z \geq 5$, where $z$ is a complex number.

Q: Can the domain of a function be a vector?

A: Yes, the domain of a function can be a vector. For example, the function $f(\mathbf{x}) = \sqrt{\mathbf{x}-5}$ has a domain of $\mathbf{x} \geq 5$, where $\mathbf{x}$ is a vector.

Q: Can the domain of a function be a matrix?

A: Yes, the domain of a function can be a matrix. For example, the function $f(\mathbf{A}) = \sqrt{\mathbf{A}-5}$ has a domain of $\mathbf{A} \geq 5$, where $\mathbf{A}$ is a matrix.

Q: Can the domain of a function be a set?

A: Yes, the domain of a function can be a set. For example, the function $f(S) = \sqrt{S-5}$ has a domain of $S \geq 5$, where $S$ is a set.