Write The Domain In Interval Notation.19) Z ( A ) = A + 2 Z(a)=\sqrt{a+2} Z ( A ) = A + 2 ​ Domain: $\qquad$

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, it's essential to consider the domain carefully to ensure that the expression under the square root is non-negative. In this article, we will explore the domain of the function $z(a)=\sqrt{a+2}$ and express it in interval notation.

The Function $z(a)=\sqrt{a+2}$

The given function is $z(a)=\sqrt{a+2}$. To find the domain of this function, we need to determine the values of $a$ for which the expression under the square root, $a+2$, is non-negative.

Non-Negativity of $a+2$

For the expression $a+2$ to be non-negative, we must have:

$$a+2 \geq 0$$

Subtracting 2 from both sides of the inequality, we get:

$$a \geq -2$$

This means that the expression under the square root, $a+2$, is non-negative when $a$ is greater than or equal to $-2$.

Domain of the Function

Based on the non-negativity condition, we can conclude that the domain of the function $z(a)=\sqrt{a+2}$ is the set of all real numbers $a$ such that $a \geq -2$.

Interval Notation

To express the domain in interval notation, we use the following notation:

$$[-2, \infty)$$

This notation indicates that the domain of the function $z(a)=\sqrt{a+2}$ is the interval from $-2$ to infinity, including $-2$.

Conclusion

In conclusion, the domain of the function $z(a)=\sqrt{a+2}$ is the set of all real numbers $a$ such that $a \geq -2$. This can be expressed in interval notation as $[-2, \infty)$. Understanding the domain of a function is crucial in mathematics, and it's essential to consider the non-negativity condition when dealing with square root functions.

Example Applications

The domain of a function has various applications in mathematics and real-world problems. For instance:

• In calculus, the domain of a function is used to determine the intervals of increase and decrease of the function.
• In optimization problems, the domain of a function is used to find the maximum or minimum value of the function.
• In statistics, the domain of a function is used to determine the range of values for which the function is defined.

Common Mistakes

When dealing with square root functions, it's common to make mistakes when determining the domain. Some common mistakes include:

• Failing to consider the non-negativity condition.
• Not expressing the domain in interval notation.
• Assuming that the domain is the entire set of real numbers.

Tips and Tricks

To avoid common mistakes when determining the domain of a square root function, follow these tips and tricks:

• Always consider the non-negativity condition.
• Express the domain in interval notation.
• Use the correct notation to indicate the domain.

Conclusion

$$$$$$$$

Introduction

In our previous article, we explored the domain of the function $z(a)=\sqrt{a+2}$ and expressed it in interval notation. In this article, we will answer some frequently asked questions about the domain of a square root function.

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

A: The domain of a square root function is the set of all real numbers for which the expression under the square root is non-negative.

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

A: To determine the domain of a square root function, you need to consider the non-negativity condition. Set the expression under the square root greater than or equal to zero and solve for the variable.

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

A: The interval notation for the domain of a square root function is $[a, \infty)$, where $a$ is the value that makes the expression under the square root equal to zero.

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

A: Yes, the domain of a square root function can be a single point. For example, if the expression under the square root is equal to zero, then the domain is a single point.

Q: Can the domain of a square root function be an empty set?

A: No, the domain of a square root function cannot be an empty set. The expression under the square root must be non-negative, so there must be at least one value that satisfies the condition.

Q: How do I express the domain of a square root function in interval notation?

A: To express the domain of a square root function in interval notation, you need to determine the value that makes the expression under the square root equal to zero and then use the interval notation $[a, \infty)$.

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, while the range of a function is the set of all possible output values.

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 one-to-one correspondence, then the domain and range are the same.

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

A: To find the domain and range of a function, you need to consider the input values and output values of the function. The domain is the set of all possible input values, while the range is the set of all possible output values.

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

A: The domain and range of a function are important because they determine the behavior of the function. The domain and range can help you understand the function's properties, such as its maximum and minimum values, its intervals of increase and decrease, and its asymptotes.

Conclusion

In conclusion, the domain of a square root function is the set of all real numbers for which the expression under the square root is non-negative. The interval notation for the domain of a square root function is $[a, \infty)$, where $a$ is the value that makes the expression under the square root equal to zero. Understanding the domain and range of a function is crucial in mathematics and has various applications in real-world problems.

Example Problems

1. Find the domain of the function $f(x)=\sqrt{x-1}$.
2. Express the domain of the function $g(x)=\sqrt{x+3}$ in interval notation.
3. Find the range of the function $h(x)=\sqrt{x-2}$.
4. Determine the domain and range of the function $j(x)=\sqrt{x+1}$.
5. Express the domain of the function $k(x)=\sqrt{x-4}$ in interval notation.

Answers

1. The domain of the function $f(x)=\sqrt{x-1}$ is $[1, \infty)$.
2. The domain of the function $g(x)=\sqrt{x+3}$ is $[-3, \infty)$.
3. The range of the function $h(x)=\sqrt{x-2}$ is $[\sqrt{2}, \infty)$.
4. The domain and range of the function $j(x)=\sqrt{x+1}$ are both $[-1, \infty)$.
5. The domain of the function $k(x)=\sqrt{x-4}$ is $[4, \infty)$.

Conclusion

In conclusion, the domain of a square root function is the set of all real numbers for which the expression under the square root is non-negative. The interval notation for the domain of a square root function is $[a, \infty)$, where $a$ is the value that makes the expression under the square root equal to zero. Understanding the domain and range of a function is crucial in mathematics and has various applications in real-world problems.