What Is ( F G ) ( X \left(\frac{f}{g}\right)(x ( G F ​ ) ( X ]?Given: F ( X ) = − X 2 − X F(x) = -x^2 - X F ( X ) = − X 2 − X G ( X ) = − 4 X + 2 G(x) = -4x + 2 G ( X ) = − 4 X + 2 Write Your Answer As A Polynomial Or A Rational Function In Simplest Form.____ , X ≠ X \neq X  = ____

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Introduction

In mathematics, the concept of a rational function is a fundamental aspect of algebra and calculus. A rational function is defined as the ratio of two polynomials, where the numerator and denominator are both polynomials. In this article, we will explore the concept of a rational function and how to simplify it. We will also provide a step-by-step guide on how to find the value of (fg)(x)\left(\frac{f}{g}\right)(x) given the functions f(x)=−x2−xf(x) = -x^2 - x and g(x)=−4x+2g(x) = -4x + 2.

What is a Rational Function?

A rational function is a function that can be expressed as the ratio of two polynomials. It is denoted as p(x)q(x)\frac{p(x)}{q(x)}, where p(x)p(x) and q(x)q(x) are polynomials. The rational function is said to be in its simplest form if the numerator and denominator have no common factors other than 1.

Simplifying a Rational Function

To simplify a rational function, we need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest polynomial that divides both the numerator and denominator. Once we have found the GCD, we can divide both the numerator and denominator by the GCD to simplify the rational function.

Finding the Value of (fg)(x)\left(\frac{f}{g}\right)(x)

Given the functions f(x)=−x2−xf(x) = -x^2 - x and g(x)=−4x+2g(x) = -4x + 2, we can find the value of (fg)(x)\left(\frac{f}{g}\right)(x) by substituting the values of f(x)f(x) and g(x)g(x) into the rational function.

Step 1: Substitute the values of f(x)f(x) and g(x)g(x) into the rational function

(fg)(x)=−x2−x−4x+2\left(\frac{f}{g}\right)(x) = \frac{-x^2 - x}{-4x + 2}

Step 2: Simplify the rational function

To simplify the rational function, we need to find the GCD of the numerator and denominator. The GCD of −x2−x-x^2 - x and −4x+2-4x + 2 is 1.

Step 3: Divide both the numerator and denominator by the GCD

(fg)(x)=−x2−x−4x+2=−x(x+1)−4x+2\left(\frac{f}{g}\right)(x) = \frac{-x^2 - x}{-4x + 2} = \frac{-x(x + 1)}{-4x + 2}

Step 4: Factor the numerator and denominator

(fg)(x)=−x(x+1)−4x+2=−x(x+1)−2(2x−1)\left(\frac{f}{g}\right)(x) = \frac{-x(x + 1)}{-4x + 2} = \frac{-x(x + 1)}{-2(2x - 1)}

Step 5: Cancel out any common factors

(fg)(x)=−x(x+1)−2(2x−1)=x(x+1)2(2x−1)\left(\frac{f}{g}\right)(x) = \frac{-x(x + 1)}{-2(2x - 1)} = \frac{x(x + 1)}{2(2x - 1)}

Step 6: Write the final answer

(fg)(x)=x(x+1)2(2x−1)\left(\frac{f}{g}\right)(x) = \frac{x(x + 1)}{2(2x - 1)}

Conclusion

In this article, we have explored the concept of a rational function and how to simplify it. We have also provided a step-by-step guide on how to find the value of (fg)(x)\left(\frac{f}{g}\right)(x) given the functions f(x)=−x2−xf(x) = -x^2 - x and g(x)=−4x+2g(x) = -4x + 2. The final answer is x(x+1)2(2x−1)\frac{x(x + 1)}{2(2x - 1)}.

Final Answer

x(x+1)2(2x−1)\boxed{\frac{x(x + 1)}{2(2x - 1)}}

Restrictions on the Domain

The domain of the rational function is all real numbers except for the values that make the denominator equal to zero. In this case, the denominator is 2(2x−1)2(2x - 1), which is equal to zero when x=12x = \frac{1}{2}. Therefore, the domain of the rational function is all real numbers except for x=12x = \frac{1}{2}.

Final Answer with Restrictions

x(x+1)2(2x−1),x≠12\boxed{\frac{x(x + 1)}{2(2x - 1)}, x \neq \frac{1}{2}}

Introduction

In our previous article, we explored the concept of a rational function and how to simplify it. We also provided a step-by-step guide on how to find the value of (fg)(x)\left(\frac{f}{g}\right)(x) given the functions f(x)=−x2−xf(x) = -x^2 - x and g(x)=−4x+2g(x) = -4x + 2. In this article, we will answer some frequently asked questions about rational functions.

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 denoted as p(x)q(x)\frac{p(x)}{q(x)}, where p(x)p(x) and q(x)q(x) are polynomials.

Q: How do I simplify a rational function?

A: To simplify a rational function, you need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest polynomial that divides both the numerator and denominator. Once you have found the GCD, you can divide both the numerator and denominator by the GCD to simplify the rational function.

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. In other words, the domain of a rational function is all real numbers except for the values that would make the function undefined.

Q: How do I find the value of a rational function at a given point?

A: To find the value of a rational function at a given point, you need to substitute the value of the point into the rational function. For example, if you want to find the value of (fg)(x)\left(\frac{f}{g}\right)(x) at x=2x = 2, you would substitute x=2x = 2 into the rational function and simplify.

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 be raised to a power that is greater than or equal to 1. For example, the rational function xx2\frac{x}{x^2} is valid, but the rational function xx\frac{x}{x} is not valid because the denominator is not raised to a power that is greater than or equal to 1.

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

A: Yes, a rational function can have a constant in the denominator. For example, the rational function x2\frac{x}{2} is valid.

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 graph the 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, where each term is a constant or a variable raised to a power.

Q: Can a rational function be a polynomial function?

A: Yes, a rational function can be a polynomial function if the denominator is a constant. For example, the rational function x1\frac{x}{1} is a polynomial function.

Q: Can a polynomial function be a rational function?

A: Yes, a polynomial function can be a rational function if the denominator is a variable raised to a power that is greater than or equal to 1. For example, the polynomial function x2x^2 is a rational function.

Conclusion

In this article, we have answered some frequently asked questions about rational functions. We have also provided a step-by-step guide on how to simplify a rational function and how to find the value of a rational function at a given point. We hope that this article has been helpful in understanding rational functions.

Final Answer

x(x+1)2(2x−1),x≠12\boxed{\frac{x(x + 1)}{2(2x - 1)}, x \neq \frac{1}{2}}