Characteristics of a Rational Function

Oct 13, 2025 by miodeya.com 29 views

Which of the Following is Written as a Rational Function?

In mathematics, a rational function is a function that can be expressed as the ratio of two polynomials. It is a fundamental concept in algebra and is used to model various real-world phenomena. In this article, we will discuss the characteristics of a rational function and identify which of the given options is written as a rational function.

A rational function is a function that can be expressed in the form:

f(x) = \frac{p(x)}{q(x)}

where $p(x)$ and $q(x)$ are polynomials, and $q(x)$ is not equal to zero. The domain of a rational function is all real numbers except for the values that make the denominator equal to zero.

Characteristics of a Rational Function

A rational function has the following characteristics:

It is a ratio of two polynomials.
The denominator is not equal to zero.
The domain is all real numbers except for the values that make the denominator equal to zero.

Examples of Rational Functions

Some examples of rational functions include:

$f(x) = \frac{x+2}{x-1}$
$g(x) = \frac{x^2-4}{x+2}$
$h(x) = \frac{3x-2}{x+1}$

Now, let's analyze the given options to determine which one is written as a rational function.

Option A: $Q(x) = x^2 - 8x + 1$

This is a quadratic function, not a rational function. It is a polynomial function, and it does not meet the criteria for a rational function.

Option B: $P(x) = 3x + 4$

This is a linear function, not a rational function. It is a polynomial function, and it does not meet the criteria for a rational function.

Option C: $F(x) = \frac{x+2}{5x}$

This is a rational function. It is a ratio of two polynomials, and the denominator is not equal to zero. Therefore, it meets the criteria for a rational function.

Option D: $G(x) = -2x$

This is a linear function, not a rational function. It is a polynomial function, and it does not meet the criteria for a rational function.

In conclusion, the correct answer is Option C: $F(x) = \frac{x+2}{5x}$ . This is the only option that meets the criteria for a rational function. A rational function is a function that can be expressed as the ratio of two polynomials, and the denominator is not equal to zero. Therefore, it is essential to identify the characteristics of a rational function to determine which of the given options is written as a rational function.

[1] "Rational Functions." Math Open Reference, mathopenref.com/rational.html.
[2] "Rational Functions." Khan Academy, khanacademy.org/math/algebra/x2f-rational-functions.

What is a rational function? A rational function is a function that can be expressed as the ratio of two polynomials.
What are the characteristics of a rational function? A rational function has the following characteristics: it is a ratio of two polynomials, the denominator is not equal to zero, and the domain is all real numbers except for the values that make the denominator equal to zero.
What are some examples of rational functions? Some examples of rational functions include $f(x) = \frac{x+2}{x-1}$ , $g(x) = \frac{x^2-4}{x+2}$ , and $h(x) = \frac{3x-2}{x+1}$ .

Rational function: A function that can be expressed as the ratio of two polynomials.
Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
Denominator: The bottom part of a fraction.
Domain: The set of all possible input values for a function.
Rational Function Q&A =====================

Q: What is a rational function?

A: A rational function is a function that can be expressed as the ratio of two polynomials. It is a fundamental concept in algebra and is used to model various real-world phenomena.

Q: What are the characteristics of a rational function?

A: A rational function has the following characteristics:

It is a ratio of two polynomials.
The denominator is not equal to zero.
The domain is all real numbers except for the values that make the denominator equal to zero.

Q: What are some examples of rational functions?

A: Some examples of rational functions include:

$f(x) = \frac{x+2}{x-1}$
$g(x) = \frac{x^2-4}{x+2}$
$h(x) = \frac{3x-2}{x+1}$

Q: How do I determine if a function is rational or not?

A: To determine if a function is rational or not, you need to check if it can be expressed as the ratio of two polynomials. If it can, then it is a rational function. If it cannot, then it is not a rational function.

Q: What is the difference between a rational function and a polynomial function?

A: A rational function is a function that can be expressed as the ratio of two polynomials, while a polynomial function is a function that can be expressed as a sum of terms, each of which is a constant or a product of a constant and a variable.

Q: Can a rational function have a variable in the denominator?

A: Yes, a rational function can have a variable in the denominator. However, the variable must not be equal to zero.

Q: How do I simplify a rational function?

A: To simplify a rational function, you need to factor the numerator and denominator, and then cancel out any common factors.

Q: What is the domain of a rational function?

A: The domain of a rational function is all real numbers except for the values that make the denominator equal to zero.

Q: Can a rational function have a constant in the denominator?

A: Yes, a rational function can have a constant in the denominator. However, the constant must not be equal to zero.

Q: How do I graph a rational function?

A: To graph a rational function, you need to find the x-intercepts, y-intercepts, and any vertical asymptotes. You can then use this information to sketch the graph of the function.

Q: What is the difference between a rational function and a trigonometric function?

A: A rational function is a function that can be expressed as the ratio of two polynomials, while a trigonometric function is a function that involves trigonometric functions such as sine, cosine, and tangent.

Q: Can a rational function be used to model real-world phenomena?

A: Yes, a rational function can be used to model real-world phenomena such as population growth, chemical reactions, and electrical circuits.

Q: How do I use a rational function to model a real-world phenomenon?

A: To use a rational function to model a real-world phenomenon, you need to identify the variables and parameters involved, and then use the function to describe the relationship between the variables.

Rational function: A function that can be expressed as the ratio of two polynomials.
Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
Denominator: The bottom part of a fraction.
Domain: The set of all possible input values for a function.
Variable: A symbol that represents a value that can change.
Parameter: A value that is used to describe a function or a relationship.
Graph: A visual representation of a function or a relationship.
Asymptote: A line that the graph of a function approaches but never touches.
Trigonometric function: A function that involves trigonometric functions such as sine, cosine, and tangent.