Given Functions M ( X ) = 1 X M(x)=\frac{1}{\sqrt{x}} M ( X ) = X ​ 1 ​ And P ( X ) = X 2 − 16 P(x)=x^2-16 P ( X ) = X 2 − 16 , State The Domains Of The Following Functions Using Interval Notation.1. Domain Of M ( X ) P ( X ) \frac{m(x)}{p(x)} P ( X ) M ( X ) ​ : $\square$2. Domain Of M ( P ( X ) M(p(x) M ( P ( X ) ]:

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 composite functions, it is essential to determine the domain of each individual function and then find the intersection of these domains to obtain the domain of the composite function. In this article, we will explore the domains of the given functions $m(x)=\frac{1}{\sqrt{x}}$ and $p(x)=x^2-16$, and then find the domains of the composite functions $\frac{m(x)}{p(x)}$ and $m(p(x))$.

Function $m(x)$

The function $m(x)$ is defined as $m(x)=\frac{1}{\sqrt{x}}$. To find the domain of this function, we need to consider the restrictions on the input values. The square root function is defined only for non-negative real numbers, so we must have $x \geq 0$. Additionally, the denominator of the function cannot be zero, so we must have $\sqrt{x} \neq 0$, which implies that $x \neq 0$. Therefore, the domain of the function $m(x)$ is the set of all non-negative real numbers except zero, which can be expressed in interval notation as $(0, \infty)$.

Function $p(x)$

The function $p(x)$ is defined as $p(x)=x^2-16$. To find the domain of this function, we need to consider the restrictions on the input values. Since the function is a polynomial, it is defined for all real numbers. However, we need to find the values of $x$ for which the function is equal to zero, as these values will be excluded from the domain. Setting $p(x)=0$, we get $x^2-16=0$, which implies that $x^2=16$. Taking the square root of both sides, we get $x=\pm 4$. Therefore, the domain of the function $p(x)$ is the set of all real numbers except $-4$ and $4$, which can be expressed in interval notation as $(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$.

Domain of $\frac{m(x)}{p(x)}$

To find the domain of the function $\frac{m(x)}{p(x)}$, we need to consider the restrictions on the input values. The function is defined only when the denominator is non-zero, so we must have $p(x) \neq 0$. From the previous section, we know that the domain of the function $p(x)$ is the set of all real numbers except $-4$ and $4$. Therefore, the domain of the function $\frac{m(x)}{p(x)}$ is the set of all real numbers except $-4$ and $4$, which can be expressed in interval notation as $(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$.

Domain of $m(p(x))$

To find the domain of the function $m(p(x))$, we need to consider the restrictions on the input values. The function $m(x)$ is defined only when the input value is non-negative, so we must have $p(x) \geq 0$. From the previous section, we know that the domain of the function $p(x)$ is the set of all real numbers except $-4$ and $4$. Therefore, the domain of the function $m(p(x))$ is the set of all real numbers except $-4$ and $4$, which can be expressed in interval notation as $(-\infty, -4) \cup (-4, 4) \cup (4, \infty)$.

Conclusion

In this article, we have explored the domains of the given functions $m(x)=\frac{1}{\sqrt{x}}$ and $p(x)=x^2-16$, and then found the domains of the composite functions $\frac{m(x)}{p(x)}$ and $m(p(x))$. We have shown that the domain of the function $\frac{m(x)}{p(x)}$ is the set of all real numbers except $-4$ and $4$, and the domain of the function $m(p(x))$ is also the set of all real numbers except $-4$ and $4$. These results demonstrate the importance of considering the restrictions on the input values when working with composite functions.

References

• [1] "Functions and Graphs" by Michael Corral, 2019.
• [2] "Calculus" by Michael Spivak, 2008.

Glossary

• Domain: The set of all possible input values for which a function is defined.
• Composite function: A function that is defined in terms of another function.
• Interval notation: A way of expressing a set of real numbers using intervals.
  Domain of Composite Functions: A Comprehensive Analysis ===========================================================

Q&A: Domain of Composite Functions

Q: What is the domain of a composite function?

A: The domain of a composite function is the set of all possible input values for which the function is defined. To find the domain of a composite function, we need to consider the restrictions on the input values of each individual function and then find the intersection of these domains.

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

A: To find the domain of a composite function, follow these steps:

1. Find the domain of each individual function.
2. Consider the restrictions on the input values of each function.
3. Find the intersection of the domains of each function.

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

A: The domain of a function is the set of all possible input values for which the function is defined. The domain of a composite function is the set of all possible input values for which the composite function is defined.

Q: Can the domain of a composite function be different from the domain of each individual function?

A: Yes, the domain of a composite function can be different from the domain of each individual function. This is because the composite function may have additional restrictions on the input values.

Q: How do I determine the domain of a composite function when one of the functions has a restricted domain?

A: To determine the domain of a composite function when one of the functions has a restricted domain, follow these steps:

1. Find the domain of the function with the restricted domain.
2. Consider the restrictions on the input values of the other function.
3. Find the intersection of the domains of each function.

Q: Can the domain of a composite function be expressed in interval notation?

A: Yes, the domain of a composite function can be expressed in interval notation. Interval notation is a way of expressing a set of real numbers using intervals.

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

A: To express the domain of a composite function in interval notation, follow these steps:

1. Find the domain of the composite function.
2. Express the domain in interval notation using the following notation:
   • $(a, b)$: the set of all real numbers between $a$ and $b$.
   • $[a, b]$ : the set of all real numbers between $a$ and $b$, including the endpoints.
   • $(-\infty, a)$: the set of all real numbers less than $a$.
   • $(a, \infty)$: the set of all real numbers greater than $a$.

Q: What are some common mistakes to avoid when finding the domain of a composite function?

A: Some common mistakes to avoid when finding the domain of a composite function include:

• Failing to consider the restrictions on the input values of each function.
• Failing to find the intersection of the domains of each function.
• Expressing the domain in interval notation incorrectly.

Q: How can I practice finding the domain of composite functions?

A: You can practice finding the domain of composite functions by working through examples and exercises. You can also use online resources and tools to help you practice.

Q: What are some real-world applications of finding the domain of composite functions?

A: Finding the domain of composite functions has many real-world applications, including:

• Calculating the area under curves.
• Finding the maximum and minimum values of functions.
• Solving optimization problems.

Conclusion

In this article, we have explored the domain of composite functions and provided answers to frequently asked questions. We have also discussed the importance of considering the restrictions on the input values when working with composite functions. By following the steps outlined in this article, you can find the domain of composite functions and apply this knowledge to real-world problems.