Find And Simplify (f G)(x) And (f-g)(x) Functions Explained

In the realm of mathematical functions, understanding how to combine and manipulate them is a fundamental skill. This article delves into the process of finding and simplifying the product and difference of two given functions, denoted as (f g)(x) and (f-g)(x), respectively. We will specifically focus on the functions f(x) = x - 4 and g(x) = x^2 - 16, and explore whether the resulting functions are the same. By examining these operations, we gain insights into the behavior and relationships between functions.

Before diving into the specifics of our functions, it's crucial to grasp the basic concepts of function operations. When we talk about (f g)(x), we're referring to the product of the two functions, which means multiplying f(x) and g(x). On the other hand, (f-g)(x) represents the difference between the two functions, obtained by subtracting g(x) from f(x). These operations are essential tools in calculus, analysis, and various other mathematical fields. They allow us to build new functions from existing ones, creating more complex models and relationships. Furthermore, simplifying these resulting functions often involves algebraic manipulations such as factoring, expanding, and combining like terms. This process not only makes the functions easier to work with but also reveals underlying structures and properties.

To find (f g)(x), we need to multiply the functions f(x) = x - 4 and g(x) = x^2 - 16. This operation combines the two functions into a single expression, revealing how their individual behaviors interact. The multiplication process is straightforward: we multiply each term of the first function by each term of the second function. This yields: (f g)(x) = (x - 4)(x^2 - 16). Expanding this product, we get x(x^2 - 16) - 4(x^2 - 16). Distributing further, we have x^3 - 16x - 4x^2 + 64. Rearranging the terms in descending order of powers, we arrive at the simplified expression: (f g)(x) = x^3 - 4x^2 - 16x + 64. This cubic function represents the product of the original two functions. Analyzing the cubic function provides valuable insights into its behavior, such as its roots, turning points, and end behavior. For instance, we can observe that the function's degree (3) indicates that it will have at most three real roots, and its end behavior will be such that as x approaches positive or negative infinity, the function will also approach infinity or negative infinity, respectively. Moreover, the coefficients of the terms reveal information about the function's shape and position on the coordinate plane. The constant term, 64, represents the y-intercept of the function, which is the point where the graph of the function intersects the y-axis. The quadratic term, -4x^2, indicates that the function has a concave-down nature, meaning it opens downward. The linear term, -16x, affects the slope and direction of the function, while the cubic term, x^3, determines the overall shape and growth rate of the function. In summary, finding (f g)(x) involves multiplying the two functions together and simplifying the resulting expression. This process leads to a new function that encapsulates the combined behavior of the original functions.

Now, let's find (f-g)(x), which means subtracting the function g(x) from f(x). Given f(x) = x - 4 and g(x) = x^2 - 16, the subtraction is performed as follows: (f-g)(x) = (x - 4) - (x^2 - 16). To simplify this, we distribute the negative sign across the terms of g(x): (f-g)(x) = x - 4 - x^2 + 16. Combining like terms, we rearrange the expression to get (f-g)(x) = -x^2 + x + 12. This quadratic function represents the difference between the original two functions. Analyzing this quadratic function provides valuable insights into its behavior, such as its roots, vertex, and concavity. The leading coefficient, -1, indicates that the parabola opens downward, meaning it has a maximum value. The vertex of the parabola, which represents the maximum point of the function, can be found using the formula x = -b / 2a, where a and b are the coefficients of the quadratic and linear terms, respectively. In this case, x = -1 / (2 * -1) = 0.5. Substituting this value into the function gives the y-coordinate of the vertex: y = -(0.5)^2 + 0.5 + 12 = 12.25. Therefore, the vertex of the parabola is at the point (0.5, 12.25). The roots of the quadratic function, which are the x-values where the function equals zero, can be found by solving the equation -x^2 + x + 12 = 0. This can be done by factoring the quadratic expression, using the quadratic formula, or completing the square. Factoring the quadratic expression gives: -(x - 4)(x + 3) = 0. Setting each factor equal to zero yields the roots x = 4 and x = -3. These roots represent the x-intercepts of the parabola, which are the points where the graph of the function intersects the x-axis. In summary, finding (f-g)(x) involves subtracting g(x) from f(x) and simplifying the resulting expression. This process leads to a new quadratic function that encapsulates the difference in behavior between the original functions.

To determine if (f g)(x) and (f-g)(x) are the same function, we compare their simplified expressions. We found that (f g)(x) = x^3 - 4x^2 - 16x + 64, which is a cubic function. On the other hand, (f-g)(x) = -x^2 + x + 12, which is a quadratic function. Since a cubic function and a quadratic function have different degrees and behaviors, they are fundamentally different types of functions. Therefore, (f g)(x) and (f-g)(x) are not the same function. The key difference lies in their degrees: the cubic function has a degree of 3, while the quadratic function has a degree of 2. This difference in degree leads to significant variations in their graphs, roots, turning points, and end behaviors. For instance, a cubic function can have up to three real roots, while a quadratic function can have at most two real roots. Similarly, a cubic function can have up to two turning points (local maxima and minima), while a quadratic function has only one turning point (its vertex). Moreover, the end behaviors of the two functions differ significantly. As x approaches positive or negative infinity, a cubic function will approach positive or negative infinity, depending on the sign of its leading coefficient. In contrast, as x approaches positive or negative infinity, a quadratic function will approach either positive or negative infinity, depending on the sign of its leading coefficient and the concavity of its graph. Therefore, based on their differing expressions and fundamental characteristics, we can definitively conclude that (f g)(x) and (f-g)(x) are distinct functions. This underscores the importance of understanding function operations and their effects on the resulting functions. By performing operations such as multiplication and subtraction, we can create new functions with unique properties and behaviors, which may differ significantly from the original functions.

In conclusion, we have successfully found and simplified the functions (f g)(x) and (f-g)(x) for the given functions f(x) = x - 4 and g(x) = x^2 - 16. We determined that (f g)(x) = x^3 - 4x^2 - 16x + 64 and (f-g)(x) = -x^2 + x + 12. Through comparison, we confirmed that these functions are not the same due to their different degrees and behaviors. This exercise highlights the importance of understanding function operations and their implications in mathematics. The ability to manipulate and simplify functions is crucial for solving complex problems and gaining deeper insights into mathematical relationships. By mastering these skills, we can unlock the full potential of mathematical functions and apply them effectively in various fields, including science, engineering, and economics. Furthermore, this analysis emphasizes the significance of distinguishing between different types of functions, such as cubic and quadratic functions, based on their degrees and properties. Each type of function exhibits unique characteristics that govern its behavior and applications. By recognizing these differences, we can choose the appropriate functions to model real-world phenomena and make accurate predictions. Ultimately, the concepts and techniques explored in this article contribute to a stronger foundation in mathematics and enhance our ability to tackle challenging mathematical problems. As we continue to explore the world of functions, we will encounter even more complex operations and relationships, which will further enrich our mathematical understanding and problem-solving skills.