Find The Domain Of The Given Function. Enter The Solution Using Interval Notation.${ N(x) = \sqrt{x - 15} }$

Introduction

In mathematics, the domain of a function is the set of all possible input values for which the function is defined. When dealing with square root functions, we need to consider the non-negativity of the expression inside the square root. In this article, we will explore how to find the domain of the given function $n(x) = \sqrt{x - 15}$ using interval notation.

Understanding the Square Root Function

The square root function is defined as $\sqrt{x} = y$, where $y$ is the value that, when multiplied by itself, gives $x$. In other words, $\sqrt{x} = y$ if and only if $y^2 = x$. For the given function $n(x) = \sqrt{x - 15}$, we need to find the values of $x$ for which the expression inside the square root is non-negative.

The Domain of the Square Root Function

To find the domain of the square root function, we need to consider the following conditions:

• The expression inside the square root must be non-negative, i.e., $x - 15 \geq 0$.
• The expression inside the square root must be real, i.e., $x - 15 \geq 0$.

Solving the inequality $x - 15 \geq 0$, we get:

$x \geq 15$

This means that the domain of the square root function is all real numbers greater than or equal to 15.

Interval Notation

Interval notation is a way of representing a set of numbers using intervals. For example, the set of all real numbers greater than or equal to 15 can be represented as $[15, \infty)$.

Conclusion

In conclusion, the domain of the given function $n(x) = \sqrt{x - 15}$ is all real numbers greater than or equal to 15, represented in interval notation as $[15, \infty)$.

Example

Let's consider an example to illustrate the concept of finding the domain of a square root function.

Suppose we have the function $f(x) = \sqrt{x + 2}$. To find the domain of this function, we need to consider the non-negativity of the expression inside the square root.

The expression inside the square root is $x + 2$, which must be non-negative. Solving the inequality $x + 2 \geq 0$, we get:

$x \geq -2$

This means that the domain of the function $f(x) = \sqrt{x + 2}$ is all real numbers greater than or equal to -2, represented in interval notation as $[-2, \infty)$.

Tips and Tricks

Here are some tips and tricks to help you find the domain of a square root function:

• Always check the non-negativity of the expression inside the square root.
• Use interval notation to represent the domain of the function.
• Consider the real numbers greater than or equal to a certain value as the domain of the function.

Common Mistakes

Here are some common mistakes to avoid when finding the domain of a square root function:

• Not checking the non-negativity of the expression inside the square root.
• Not using interval notation to represent the domain of the function.
• Considering the wrong set of numbers as the domain of the function.

Real-World Applications

The concept of finding the domain of a square root function has many real-world applications. For example:

• In physics, the domain of a function can represent the range of possible values for a physical quantity.
• In engineering, the domain of a function can represent the range of possible values for a design parameter.
• In economics, the domain of a function can represent the range of possible values for a economic variable.

Conclusion

Q: What is the domain of a square root function?

A: The domain of a square root function is the set of all possible input values for which the function is defined. In other words, it is the set of all real numbers for which the expression inside the square root is non-negative.

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

A: To find the domain of a square root function, you need to consider the non-negativity of the expression inside the square root. You can do this by solving the inequality $x - a \geq 0$, where $a$ is the constant term inside the square root.

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

A: The domain of a function is the set of all possible input values for which the function is defined, while the range of a function is the set of all possible output values. In other words, the domain is the set of all possible $x$ values, while the range is the set of all possible $y$ values.

Q: Can the domain of a function be empty?

A: Yes, the domain of a function can be empty. This occurs when the expression inside the square root is always negative, or when the function is undefined for all possible input values.

Q: How do I represent the domain of a function in interval notation?

A: To represent the domain of a function in interval notation, you need to use the following notation:

• $[a, b)$ represents the set of all real numbers greater than or equal to $a$ and less than $b$.
• $(a, b]$ represents the set of all real numbers greater than $a$ and less than or equal to $b$.
• $[a, \infty)$ represents the set of all real numbers greater than or equal to $a$.
• $(-\infty, a]$ represents the set of all real numbers less than or equal to $a$.

Q: What is the domain of the function $f(x) = \sqrt{x + 3}$?

A: To find the domain of the function $f(x) = \sqrt{x + 3}$, we need to consider the non-negativity of the expression inside the square root. Solving the inequality $x + 3 \geq 0$, we get:

$x \geq -3$

This means that the domain of the function $f(x) = \sqrt{x + 3}$ is all real numbers greater than or equal to -3, represented in interval notation as $[-3, \infty)$.

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

A: To find the domain of the function $g(x) = \sqrt{x - 2}$, we need to consider the non-negativity of the expression inside the square root. Solving the inequality $x - 2 \geq 0$, we get:

$x \geq 2$

This means that the domain of the function $g(x) = \sqrt{x - 2}$ is all real numbers greater than or equal to 2, represented in interval notation as $[2, \infty)$.

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

A: Yes, the domain of a function can be a single point. This occurs when the expression inside the square root is equal to zero, or when the function is undefined for all possible input values except for one.

Q: How do I find the domain of a function with a square root in the denominator?

A: To find the domain of a function with a square root in the denominator, you need to consider the non-negativity of the expression inside the square root. You can do this by solving the inequality $x - a \geq 0$, where $a$ is the constant term inside the square root.

Q: What is the domain of the function $h(x) = \frac{1}{\sqrt{x - 1}}$?

A: To find the domain of the function $h(x) = \frac{1}{\sqrt{x - 1}}$, we need to consider the non-negativity of the expression inside the square root. Solving the inequality $x - 1 \geq 0$, we get:

$x \geq 1$

This means that the domain of the function $h(x) = \frac{1}{\sqrt{x - 1}}$ is all real numbers greater than or equal to 1, represented in interval notation as $[1, \infty)$.

Conclusion

In conclusion, finding the domain of a square root function is an important concept in mathematics. By understanding the non-negativity of the expression inside the square root and using interval notation, we can find the domain of a function and represent it in a clear and concise manner.