Finding Points On Rational Function Graph F(x) = (5x + 14) / (6x^2 + X - 2) Where F(x) = 1

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Introduction

In the realm of mathematical functions, rational functions hold a significant place, characterized by their unique properties and wide-ranging applications. Understanding the behavior of rational functions is crucial in various fields, including calculus, algebra, and real-world modeling. This article delves into the process of finding a specific point on the graph of a given rational function where the function value equals 1. We will focus on the rational function f(x) = (5x + 14) / (6x^2 + x - 2) and explore the steps involved in determining the coordinates of the point where f(x) = 1. This exploration will not only enhance our understanding of rational functions but also provide a practical approach to solving similar problems.

Understanding Rational Functions

Before we dive into the specifics of our problem, let's briefly revisit the concept of rational functions. A rational function is defined as a function that can be expressed as the quotient of two polynomials. In other words, it is a function of the form f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomial functions, and Q(x) is not equal to zero. The denominator, Q(x), plays a critical role in defining the function's domain, as it cannot be zero. Zeros of the denominator correspond to vertical asymptotes or holes in the graph of the rational function. The function we are dealing with, f(x) = (5x + 14) / (6x^2 + x - 2), perfectly fits this definition, making it a prime example of a rational function. Analyzing rational functions often involves identifying their asymptotes, intercepts, and general behavior, which helps in sketching their graphs and understanding their properties. This foundation is essential for solving problems related to rational functions, including finding specific points on their graphs.

Problem Statement: Finding the Point Where f(x) = 1

The core of our discussion lies in identifying the point on the graph of the rational function f(x) = (5x + 14) / (6x^2 + x - 2) where the function value, f(x), equals 1. This task requires us to solve the equation (5x + 14) / (6x^2 + x - 2) = 1 for x. The solutions for x will then be used to find the corresponding y-coordinates (which are equal to 1 in this case), giving us the coordinates of the point(s) where the graph of the function intersects the horizontal line y = 1. This problem combines algebraic manipulation with an understanding of function behavior. By solving this equation, we gain insight into the specific points on the graph that satisfy the given condition, which is a fundamental concept in the study of functions and their graphical representations.

Step-by-Step Solution

To find the point on the graph where f(x) = 1, we need to solve the equation (5x + 14) / (6x^2 + x - 2) = 1. This involves several algebraic steps, which we will break down for clarity:

  1. Set up the equation: Our starting point is the equation (5x + 14) / (6x^2 + x - 2) = 1. This equation represents the condition we want to satisfy: the function value equals 1.
  2. Multiply both sides by the denominator: To eliminate the fraction, we multiply both sides of the equation by the denominator, 6x^2 + x - 2. This gives us 5x + 14 = 6x^2 + x - 2. This step is crucial as it transforms the rational equation into a more manageable polynomial equation.
  3. Rearrange the equation into a quadratic form: We need to rearrange the equation to the standard quadratic form, which is ax^2 + bx + c = 0. Subtracting 5x and 14 from both sides, we get 0 = 6x^2 + x - 5x - 2 - 14, which simplifies to 6x^2 - 4x - 16 = 0. This form allows us to apply standard methods for solving quadratic equations.
  4. Simplify the quadratic equation: To make the equation easier to work with, we can simplify it by dividing all terms by their greatest common divisor, which is 2. This simplifies the equation to 3x^2 - 2x - 8 = 0. Simplified equations are generally easier to solve and less prone to errors.
  5. Solve the quadratic equation: We can solve the quadratic equation 3x^2 - 2x - 8 = 0 using factoring, completing the square, or the quadratic formula. In this case, factoring is a straightforward approach. We look for two numbers that multiply to (3)(-8) = -24 and add up to -2. These numbers are -6 and 4. So, we rewrite the middle term as -6x + 4x: 3x^2 - 6x + 4x - 8 = 0. Now, we factor by grouping: 3x(x - 2) + 4(x - 2) = 0, which gives us (3x + 4)(x - 2) = 0. This step is a critical skill in algebra, and mastering it is essential for solving many types of equations.
  6. Find the values of x: Setting each factor equal to zero, we get 3x + 4 = 0 and x - 2 = 0. Solving these equations gives us x = -4/3 and x = 2. These are the x-coordinates of the points where the function's value is 1.
  7. Find the corresponding y-coordinates: Since we are looking for points where f(x) = 1, the y-coordinate for both solutions is 1. Therefore, the points are (-4/3, 1) and (2, 1). These points are the intersection of the graph of the rational function and the horizontal line y = 1.
  8. Verify the solutions: It's always a good practice to verify the solutions by plugging them back into the original equation. For x = -4/3, f(-4/3) = (5(-4/3) + 14) / (6(-4/3)^2 + (-4/3) - 2) = 1. For x = 2, f(2) = (5(2) + 14) / (6(2)^2 + 2 - 2) = 1. Both solutions satisfy the original equation, confirming our results.

By following these steps, we have successfully found the points on the graph of the rational function f(x) where the function value is equal to 1. This process highlights the importance of algebraic manipulation and problem-solving skills in understanding the behavior of functions.

Graphical Interpretation

Having found the points (-4/3, 1) and (2, 1) where f(x) = 1, it's beneficial to visualize this result graphically. The graph of the rational function f(x) = (5x + 14) / (6x^2 + x - 2) is a curve with certain characteristic features, such as vertical asymptotes where the denominator is zero and horizontal asymptotes determined by the degrees of the numerator and denominator. The points we found represent the intersections of this curve with the horizontal line y = 1. Graphically, these points are where the curve crosses or touches the line y = 1. Visualizing the graph helps to confirm our algebraic solutions and provides a deeper understanding of the function's behavior. For instance, we can see how the function approaches the value 1 at these specific x-coordinates and how it behaves around these points. This graphical interpretation is a powerful tool in understanding the properties and characteristics of rational functions.

Conclusion

In this article, we successfully found the points on the graph of the rational function f(x) = (5x + 14) / (6x^2 + x - 2) where f(x) = 1. By setting up the equation (5x + 14) / (6x^2 + x - 2) = 1 and systematically solving it, we identified the x-coordinates as -4/3 and 2. These values correspond to the points (-4/3, 1) and (2, 1) on the graph. This exercise demonstrates the application of algebraic techniques to solve problems involving rational functions. It also highlights the importance of understanding the properties of rational functions, such as their domain and graphical behavior. The ability to find specific points on the graph of a function is a fundamental skill in mathematics, with applications in various fields. This problem serves as a valuable example of how to approach such tasks and reinforces the connection between algebraic solutions and graphical representations. Furthermore, the process of solving this problem enhances our understanding of rational functions and their role in mathematical analysis.