Given F(x) = X^2 - 4 And G(x) = -7x + 3, What Is (f/g)(3/7), If It Exists?

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Introduction

In this comprehensive article, we will delve into the process of evaluating the composite function (f/g)(3/7), given the functions f(x) = x^2 - 4 and g(x) = -7x + 3. This problem combines concepts from basic algebra, function evaluation, and function composition, offering a great exercise in understanding how functions interact. The primary goal is to find the value of the function obtained by dividing f(x) by g(x) and then substituting x = 3/7 into the resulting expression. This task involves several steps, including defining the composite function (f/g)(x), simplifying it if possible, and then evaluating it at the specified point. We will also address the critical aspect of ensuring that the function is defined at the given point, which means checking that the denominator, g(x), is not zero when x = 3/7. This exploration will not only provide a step-by-step solution to the specific problem but also enhance understanding of the broader concepts of function composition and evaluation. The knowledge gained here can be applied to various mathematical contexts, making this a valuable exercise for students and anyone interested in mathematics. By the end of this discussion, you should have a clear understanding of how to approach similar problems involving composite functions and function evaluation.

Defining the Functions

To begin, let's clearly define the given functions. We have f(x) = x^2 - 4, which is a quadratic function. This function takes an input x, squares it, and then subtracts 4 from the result. Quadratic functions are characterized by their parabolic shape when graphed and are fundamental in algebra and calculus. The second function is g(x) = -7x + 3, which is a linear function. Linear functions have a constant rate of change and form a straight line when graphed. Understanding the nature of these functions is crucial for performing operations on them and evaluating them at specific points. The function f(x) represents a parabola that opens upwards, with its vertex at (0, -4). The roots of f(x), where f(x) = 0, are x = ±2. These roots are important as they indicate where the function crosses the x-axis. The function g(x), on the other hand, is a straight line with a negative slope, meaning it decreases as x increases. The y-intercept of g(x) is 3, and the x-intercept, where g(x) = 0, is x = 3/7. This x-intercept is particularly relevant to our problem, as it is the point at which the denominator of the composite function (f/g)(x) could be zero, which would make the function undefined. Therefore, before we proceed with evaluating (f/g)(3/7), we must ensure that g(3/7) is not equal to zero. This preliminary check is essential to avoid mathematical errors and ensure the validity of our solution. Understanding these individual functions is the foundation for understanding their combined behavior when we form the composite function (f/g)(x).

Constructing (f/g)(x)

Now, let's construct the composite function (f/g)(x). This notation means we are dividing the function f(x) by the function g(x). So, (f/g)(x) = f(x) / g(x). Substituting the given functions, we have (f/g)(x) = (x^2 - 4) / (-7x + 3). This composite function represents a rational function, which is a function formed by dividing one polynomial by another. Rational functions can have interesting behaviors, such as vertical asymptotes where the denominator is zero. Understanding the structure of this composite function is key to evaluating it correctly. Before proceeding, it's beneficial to consider if the expression can be simplified further. The numerator, x^2 - 4, is a difference of squares and can be factored as (x - 2)(x + 2). However, the denominator, -7x + 3, cannot be factored easily and doesn't share any common factors with the numerator. Therefore, the expression (x^2 - 4) / (-7x + 3) is already in its simplest form. This simplification check is a good practice in general, as it can often make subsequent calculations easier. Now that we have the composite function (f/g)(x) in its simplest form, we can proceed to the next step, which is evaluating this function at x = 3/7. This will involve substituting 3/7 for x in the expression and performing the necessary arithmetic operations. However, as mentioned earlier, we must first verify that the denominator is not zero at x = 3/7 to ensure the function is defined at this point. This careful approach is essential for obtaining the correct result and avoiding potential pitfalls in mathematical calculations.

Evaluating (f/g)(3/7)

To evaluate (f/g)(3/7), we need to substitute x = 3/7 into the composite function we derived earlier, which is (f/g)(x) = (x^2 - 4) / (-7x + 3). This involves replacing every instance of x in the expression with 3/7. So, we have (f/g)(3/7) = ((3/7)^2 - 4) / (-7(3/7) + 3). Now, we perform the arithmetic operations step by step. First, let's calculate (3/7)^2, which is (3/7) * (3/7) = 9/49. Next, we calculate -7(3/7), which simplifies to -3. Now, substitute these values back into the expression: (f/g)(3/7) = (9/49 - 4) / (-3 + 3). Notice that the denominator is -3 + 3, which equals 0. This means we have a division by zero, which is undefined in mathematics. Therefore, the function (f/g)(x) is not defined at x = 3/7. This result highlights the importance of checking the denominator before proceeding with the evaluation of a rational function. If the denominator is zero at the point of evaluation, the function is undefined at that point, and there is no numerical value for the function at that x-value. In this case, (f/g)(3/7) is undefined because the denominator, -7x + 3, becomes zero when x = 3/7. This conclusion is a critical part of the solution and demonstrates a key aspect of working with rational functions. We have successfully determined that (f/g)(3/7) does not exist due to the division by zero.

Conclusion

In conclusion, we set out to find the value of (f/g)(3/7) given the functions f(x) = x^2 - 4 and g(x) = -7x + 3. We began by defining the functions and constructing the composite function (f/g)(x), which is (x^2 - 4) / (-7x + 3). We then attempted to evaluate this composite function at x = 3/7. Upon substitution, we found that the denominator, -7(3/7) + 3, equals 0. This led us to the crucial realization that division by zero is undefined in mathematics. Therefore, the value of (f/g)(3/7) does not exist. This exercise underscores the importance of checking for potential division by zero when working with rational functions. It is a fundamental principle in mathematics that division by zero is undefined, and failing to recognize this can lead to incorrect results. The steps we followed – defining the functions, constructing the composite function, substituting the value, and checking the denominator – are essential for solving similar problems. This problem serves as a valuable example for understanding the behavior of rational functions and the significance of the denominator. The process of finding (f/g)(3/7) not only provided a specific answer but also reinforced the broader concepts of function composition, function evaluation, and the domain of rational functions. Understanding these concepts is crucial for success in algebra and calculus, making this a worthwhile mathematical exploration.