Domain of the Given Function

Oct 13, 2025 by miodeya.com 131 views

Introduction

Polynomial functions are a fundamental concept in algebra, and understanding their domain is crucial for solving various mathematical problems. In this article, we will explore the domain of a given polynomial function, $r(x) = -14x^{30} + 27$ . We will analyze the function, identify its characteristics, and determine its domain using interval notation.

What is a Polynomial Function?

A polynomial function is a function that can be written in the form:

f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0

where $a_n, a_{n-1}, \ldots, a_1, a_0$ are constants, and $n$ is a non-negative integer. The domain of a polynomial function is the set of all real numbers, denoted by $\mathbb{R}$ .

The Given Function

The given function is $r(x) = -14x^{30} + 27$ . At first glance, this function appears to be a polynomial function, as it can be written in the form of a polynomial function. However, we need to analyze it further to confirm its nature.

Is the Given Function a Polynomial Function?

To determine if the given function is a polynomial function, we need to check if it meets the definition of a polynomial function. The function $r(x) = -14x^{30} + 27$ can be written as:

r(x) = -14x^{30} + 27 = -14x^{30} + 0x^{29} + 0x^{28} + \cdots + 0x + 27

This function has a non-zero constant term, $27$ , and all the other coefficients are zero. Therefore, it meets the definition of a polynomial function.

Domain of the Given Function

Since the given function is a polynomial function, its domain is the set of all real numbers, denoted by $\mathbb{R}$ . In interval notation, the domain of the given function is:

(-\infty, \infty)

Conclusion

In conclusion, the given function, $r(x) = -14x^{30} + 27$ , is a polynomial function. Its domain is the set of all real numbers, denoted by $\mathbb{R}$ , and can be represented in interval notation as $(-\infty, \infty)$ .

References

[1] "Polynomial Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/polynomial.html
[2] "Domain of a Function" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/algebra-domains

Additional Resources

[1] "Polynomial Functions" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/PolynomialFunction.html
[2] "Domain of a Function" by Brilliant. Retrieved from https://brilliant.org/wiki/domain-of-a-function/
Domain of a Polynomial Function: Q&A =====================================

Introduction

In our previous article, we explored the domain of a given polynomial function, $r(x) = -14x^{30} + 27$ . We determined that the function is a polynomial function and its domain is the set of all real numbers, denoted by $\mathbb{R}$ . In this article, we will answer some frequently asked questions related to the domain of polynomial functions.

Q: What is the domain of a polynomial function?

A: The domain of a polynomial function is the set of all real numbers, denoted by $\mathbb{R}$ . This means that the function is defined for all real values of $x$ .

Q: Can a polynomial function have a restricted domain?

A: No, a polynomial function cannot have a restricted domain. By definition, a polynomial function is defined for all real values of $x$ . However, it's worth noting that a polynomial function can have a restricted range, which is the set of all possible output values.

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

A: To determine the domain of a polynomial function, you can simply look at the function and see if it has any restrictions on the values of $x$ . If the function is a polynomial function, its domain is the set of all real numbers.

Q: Can a polynomial function have a domain that is not a subset of the real numbers?

A: No, a polynomial function cannot have a domain that is not a subset of the real numbers. By definition, a polynomial function is defined for all real values of $x$ .

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

A: The domain of a polynomial function is the set of all possible input values, while the range is the set of all possible output values. In other words, the domain is the set of all values of $x$ that the function can take, while the range is the set of all values that the function can produce.

Q: Can a polynomial function have a domain that is a subset of the real numbers?

A: Yes, a polynomial function can have a domain that is a subset of the real numbers. For example, the function $f(x) = \frac{1}{x}$ has a domain of all real numbers except $x=0$ , which is a subset of the real numbers.

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

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

If the domain is the set of all real numbers, you can represent it as $(-\infty, \infty)$ .
If the domain is a subset of the real numbers, you can represent it as an interval, such as $[a, b]$ or $(a, b)$ .

Conclusion

In conclusion, the domain of a polynomial function is the set of all real numbers, denoted by $\mathbb{R}$ . We hope that this Q&A article has helped to clarify any questions you may have had about the domain of polynomial functions.

References

[1] "Polynomial Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/polynomial.html
[2] "Domain of a Function" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/algebra-domains

Additional Resources

[1] "Polynomial Functions" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/PolynomialFunction.html
[2] "Domain of a Function" by Brilliant. Retrieved from https://brilliant.org/wiki/domain-of-a-function/