Exercise 2Consider The Function { F $}$ Defined By: ${ F(x) = \frac{-x^2 + X + 6}{x + 1} }$and Designate By (C) Its Representative Curve In An Orthonormal System.1. Justify That ${ F(x) = -x + 2 + \frac{4}{x+1} }$2.

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Introduction

In this exercise, we are given a rational function defined by f(x)=−x2+x+6x+1f(x) = \frac{-x^2 + x + 6}{x + 1}. Our objective is to simplify this function and represent its curve in an orthonormal system. To achieve this, we will first simplify the given function using algebraic manipulations and then analyze its behavior to determine its representative curve.

Simplifying the Rational Function

To simplify the given function, we can start by factoring the numerator and denominator. However, in this case, we can use a different approach to simplify the function.

We can rewrite the function as follows:

f(x)=−x2+x+6x+1f(x) = \frac{-x^2 + x + 6}{x + 1}

Using the method of partial fractions, we can express the function as:

f(x)=−x+2+4x+1f(x) = -x + 2 + \frac{4}{x+1}

This simplification can be justified by performing the following steps:

  1. Multiply both sides of the equation by (x+1)(x + 1) to eliminate the fraction:

−x2+x+6=(−x+2)(x+1)+4-x^2 + x + 6 = (-x + 2)(x + 1) + 4

  1. Expand the right-hand side of the equation:

−x2+x+6=−x2−x+2+4-x^2 + x + 6 = -x^2 - x + 2 + 4

  1. Simplify the equation by combining like terms:

−x2+x+6=−x2−x+6-x^2 + x + 6 = -x^2 - x + 6

  1. Cancel out the common terms on both sides of the equation:

0=00 = 0

This shows that the original function can be simplified to f(x)=−x+2+4x+1f(x) = -x + 2 + \frac{4}{x+1}.

Representative Curve in an Orthonormal System

To represent the curve of the simplified function in an orthonormal system, we need to analyze its behavior. The function f(x)=−x+2+4x+1f(x) = -x + 2 + \frac{4}{x+1} consists of a linear term and a rational term.

The linear term −x+2-x + 2 represents a straight line with a slope of −1-1 and a y-intercept of 22. The rational term 4x+1\frac{4}{x+1} represents a hyperbola with a horizontal asymptote at y=4y = 4.

As xx approaches negative infinity, the rational term approaches 00, and the function approaches the linear term. As xx approaches positive infinity, the rational term approaches 44, and the function approaches the horizontal asymptote.

The representative curve of the function in an orthonormal system is a combination of the linear and rational terms. The curve has a vertical asymptote at x=−1x = -1 due to the rational term, and it approaches the horizontal asymptote at y=4y = 4 as xx approaches positive infinity.

Graphical Representation

To visualize the representative curve of the function, we can plot the function using a graphing tool or software. The graph will show the curve of the function, including the vertical asymptote at x=−1x = -1 and the horizontal asymptote at y=4y = 4.

Conclusion

In this exercise, we simplified the given rational function using algebraic manipulations and analyzed its behavior to determine its representative curve in an orthonormal system. The simplified function is f(x)=−x+2+4x+1f(x) = -x + 2 + \frac{4}{x+1}, and its representative curve is a combination of a linear term and a rational term. The curve has a vertical asymptote at x=−1x = -1 and approaches the horizontal asymptote at y=4y = 4 as xx approaches positive infinity.

Key Takeaways

  • The given rational function can be simplified using algebraic manipulations.
  • The simplified function is f(x)=−x+2+4x+1f(x) = -x + 2 + \frac{4}{x+1}.
  • The representative curve of the function has a vertical asymptote at x=−1x = -1 and approaches the horizontal asymptote at y=4y = 4 as xx approaches positive infinity.

Further Exploration

  • Analyze the behavior of the function as xx approaches negative infinity.
  • Determine the domain and range of the function.
  • Graph the function using a graphing tool or software to visualize its representative curve.
    Q&A: Simplifying Rational Functions and Graphical Representation ================================================================

Introduction

In our previous article, we simplified the rational function f(x)=−x2+x+6x+1f(x) = \frac{-x^2 + x + 6}{x + 1} and analyzed its behavior to determine its representative curve in an orthonormal system. In this article, we will answer some frequently asked questions related to simplifying rational functions and graphical representation.

Q: What is the purpose of simplifying rational functions?

A: The purpose of simplifying rational functions is to make them easier to work with and understand. Simplifying rational functions can help us analyze their behavior, determine their representative curves, and solve equations involving rational functions.

Q: How do I simplify a rational function?

A: To simplify a rational function, you can use algebraic manipulations such as factoring, canceling out common terms, and using the method of partial fractions. You can also use graphing tools or software to visualize the function and determine its representative curve.

Q: What is the method of partial fractions?

A: The method of partial fractions is a technique used to simplify rational functions by expressing them as a sum of simpler fractions. This method involves factoring the denominator and expressing the rational function as a sum of fractions with linear or quadratic denominators.

Q: How do I determine the representative curve of a rational function?

A: To determine the representative curve of a rational function, you need to analyze its behavior and identify its asymptotes, intercepts, and other key features. You can use graphing tools or software to visualize the function and determine its representative curve.

Q: What is a vertical asymptote?

A: A vertical asymptote is a vertical line that the graph of a function approaches but never touches. Vertical asymptotes occur when the denominator of a rational function is equal to zero.

Q: What is a horizontal asymptote?

A: A horizontal asymptote is a horizontal line that the graph of a function approaches but never touches. Horizontal asymptotes occur when the degree of the numerator is less than or equal to the degree of the denominator.

Q: How do I graph a rational function?

A: To graph a rational function, you can use graphing tools or software such as graphing calculators, computer algebra systems, or online graphing tools. You can also use algebraic manipulations to simplify the function and determine its representative curve.

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

A: Some common mistakes to avoid when simplifying rational functions include:

  • Not factoring the numerator and denominator correctly
  • Not canceling out common terms
  • Not using the method of partial fractions correctly
  • Not analyzing the behavior of the function correctly
  • Not identifying the asymptotes, intercepts, and other key features of the function correctly

Conclusion

In this article, we answered some frequently asked questions related to simplifying rational functions and graphical representation. We hope that this article has provided you with a better understanding of how to simplify rational functions and determine their representative curves.

Key Takeaways

  • Simplifying rational functions is an important step in analyzing their behavior and determining their representative curves.
  • The method of partial fractions is a useful technique for simplifying rational functions.
  • Graphing tools or software can be used to visualize the function and determine its representative curve.
  • Vertical and horizontal asymptotes are important features of rational functions that need to be identified correctly.

Further Exploration

  • Practice simplifying rational functions using algebraic manipulations and the method of partial fractions.
  • Use graphing tools or software to visualize the function and determine its representative curve.
  • Analyze the behavior of rational functions and identify their asymptotes, intercepts, and other key features.