Which Of The Following Is Written As A Rational Function?A. F ( X ) = X − 3 8 X F(x)=\frac{x-3}{8x} F ( X ) = 8 X X − 3 ​ B. Q ( X ) = X 2 + 5 X − 6 Q(x)=x^2+5x-6 Q ( X ) = X 2 + 5 X − 6 C. G ( X ) = − 5 X G(x)=-5x G ( X ) = − 5 X D. P ( X ) = 2 X + 3 P(x)=2x+3 P ( X ) = 2 X + 3

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In mathematics, a rational function is a type of function that can be expressed as the ratio of two polynomials. It is a function that can be written in the form of a fraction, where the numerator and denominator are both polynomials. Rational functions are an essential concept in algebra and are used to model various real-world phenomena.

Understanding Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The numerator and denominator of a rational function are both polynomials, and the function is defined for all values of x except those that make the denominator equal to zero. Rational functions can be written in the form of a fraction, where the numerator and denominator are both polynomials.

Characteristics of Rational Functions

Rational functions have several characteristics that distinguish them from other types of functions. Some of the key characteristics of rational functions include:

  • Numerator and Denominator: Rational functions have a numerator and a denominator, both of which are polynomials.
  • Fractional Form: Rational functions can be written in the form of a fraction, where the numerator and denominator are both polynomials.
  • Domain: Rational functions are defined for all values of x except those that make the denominator equal to zero.
  • Vertical Asymptotes: Rational functions can have vertical asymptotes, which are vertical lines that the graph of the function approaches but never touches.

Examples of Rational Functions

There are many examples of rational functions, including:

  • Linear Rational Functions: These are rational functions where the numerator and denominator are both linear polynomials. Examples include F(x)=x−38xF(x)=\frac{x-3}{8x} and G(x)=−5xG(x)=-5x.
  • Quadratic Rational Functions: These are rational functions where the numerator and denominator are both quadratic polynomials. Examples include Q(x)=x2+5x−6Q(x)=x^2+5x-6.
  • Polynomial Rational Functions: These are rational functions where the numerator and denominator are both polynomial expressions. Examples include P(x)=2x+3P(x)=2x+3.

Which of the Following is Written as a Rational Function?

Now that we have discussed the characteristics of rational functions and provided examples of rational functions, let's determine which of the following is written as a rational function.

A. F(x)=x−38xF(x)=\frac{x-3}{8x}

This function is written as a rational function because it can be expressed as the ratio of two polynomials. The numerator is the polynomial x−3x-3, and the denominator is the polynomial 8x8x. This function meets the criteria for a rational function, as it can be written in the form of a fraction, and the numerator and denominator are both polynomials.

B. Q(x)=x2+5x−6Q(x)=x^2+5x-6

This function is not written as a rational function because it is a polynomial expression, not a ratio of two polynomials. While it can be factored into (x+6)(x−1)(x+6)(x-1), it is still a polynomial expression, not a rational function.

C. G(x)=−5xG(x)=-5x

This function is not written as a rational function because it is a linear polynomial expression, not a ratio of two polynomials. While it can be written as −5x=−5x1-5x=\frac{-5x}{1}, it is still a polynomial expression, not a rational function.

D. P(x)=2x+3P(x)=2x+3

This function is not written as a rational function because it is a linear polynomial expression, not a ratio of two polynomials. While it can be written as 2x+3=2x+312x+3=\frac{2x+3}{1}, it is still a polynomial expression, not a rational function.

Conclusion

In conclusion, the only function that is written as a rational function is F(x)=x−38xF(x)=\frac{x-3}{8x}. This function meets the criteria for a rational function, as it can be written in the form of a fraction, and the numerator and denominator are both polynomials. The other functions, Q(x)=x2+5x−6Q(x)=x^2+5x-6, G(x)=−5xG(x)=-5x, and P(x)=2x+3P(x)=2x+3, are not written as rational functions because they are polynomial expressions, not ratios of two polynomials.

References

  • Algebra: A Comprehensive Introduction, Michael Artin
  • Calculus: Early Transcendentals, James Stewart
  • Rational Functions: A Guide to Understanding and Solving Rational Functions, Math Open Reference

Additional Resources

  • Rational Functions: Khan Academy
  • Rational Functions: Mathway
  • Rational Functions: Wolfram Alpha
    Rational Functions Q&A ==========================

In this article, we will answer some frequently asked questions about rational functions. Rational functions are a type of function that can be expressed as the ratio of two polynomials. They are an essential concept in algebra and are used to model various real-world phenomena.

Q: What is a rational function?

A: A rational function is a type of function that can be expressed as the ratio of two polynomials. It is a function that can be written in the form of a fraction, where the numerator and denominator are both polynomials.

Q: What are the characteristics of a rational function?

A: The characteristics of a rational function include:

  • Numerator and Denominator: Rational functions have a numerator and a denominator, both of which are polynomials.
  • Fractional Form: Rational functions can be written in the form of a fraction, where the numerator and denominator are both polynomials.
  • Domain: Rational functions are defined for all values of x except those that make the denominator equal to zero.
  • Vertical Asymptotes: Rational functions can have vertical asymptotes, which are vertical lines that the graph of the function approaches but never touches.

Q: What are some examples of rational functions?

A: Some examples of rational functions include:

  • Linear Rational Functions: These are rational functions where the numerator and denominator are both linear polynomials. Examples include F(x)=x−38xF(x)=\frac{x-3}{8x} and G(x)=−5xG(x)=-5x.
  • Quadratic Rational Functions: These are rational functions where the numerator and denominator are both quadratic polynomials. Examples include Q(x)=x2+5x−6Q(x)=x^2+5x-6.
  • Polynomial Rational Functions: These are rational functions where the numerator and denominator are both polynomial expressions. Examples include P(x)=2x+3P(x)=2x+3.

Q: How do I determine if a function is a rational function?

A: To determine if a function is a rational function, you need to check if it can be expressed as the ratio of two polynomials. If the function can be written in the form of a fraction, where the numerator and denominator are both polynomials, then it is a rational function.

Q: What are some common mistakes to avoid when working with rational functions?

A: Some common mistakes to avoid when working with rational functions include:

  • Not checking the domain: Make sure to check the domain of the function to ensure that it is defined for all values of x.
  • Not simplifying the function: Make sure to simplify the function to its simplest form to avoid unnecessary complexity.
  • Not considering vertical asymptotes: Make sure to consider vertical asymptotes when graphing the function.

Q: How do I graph a rational function?

A: To graph a rational function, you need to follow these steps:

  1. Determine the domain: Determine the domain of the function to ensure that it is defined for all values of x.
  2. Find the vertical asymptotes: Find the vertical asymptotes of the function by setting the denominator equal to zero and solving for x.
  3. Graph the function: Graph the function using a graphing calculator or by hand.
  4. Consider the behavior of the function: Consider the behavior of the function as x approaches positive and negative infinity.

Q: What are some real-world applications of rational functions?

A: Rational functions have many real-world applications, including:

  • Modeling population growth: Rational functions can be used to model population growth and decline.
  • Modeling chemical reactions: Rational functions can be used to model chemical reactions and equilibrium.
  • Modeling electrical circuits: Rational functions can be used to model electrical circuits and circuit analysis.

Conclusion

In conclusion, rational functions are a type of function that can be expressed as the ratio of two polynomials. They are an essential concept in algebra and are used to model various real-world phenomena. By understanding the characteristics and examples of rational functions, you can better navigate the world of algebra and beyond.

References

  • Algebra: A Comprehensive Introduction, Michael Artin
  • Calculus: Early Transcendentals, James Stewart
  • Rational Functions: A Guide to Understanding and Solving Rational Functions, Math Open Reference

Additional Resources

  • Rational Functions: Khan Academy
  • Rational Functions: Mathway
  • Rational Functions: Wolfram Alpha