Function Notation Expressing 9x + 3y = 12 With X As Independent Variable

In mathematics, function notation is a symbolic way to represent a function. It provides a concise and clear way to express the relationship between an input and an output. In this comprehensive guide, we will explore how to express the given linear equation, $9x + 3y = 12$, using function notation, where x is the independent variable. Understanding function notation is crucial for comprehending various mathematical concepts and their applications. A function, in simple terms, is a rule that assigns a unique output value for each input value. Function notation allows us to represent this rule in a standardized format, making it easier to analyze and manipulate mathematical relationships. The standard notation for a function is written as f(x), where f is the name of the function and x is the input variable. The expression f(x) represents the output value of the function when the input is x. Function notation provides a clear way to identify the input and output of a function, which is particularly useful when dealing with complex mathematical models. It also facilitates the representation of mathematical relationships in a concise and unambiguous manner. This notation is widely used in various branches of mathematics, including calculus, algebra, and analysis, and is essential for understanding more advanced mathematical concepts. The ability to effectively use and interpret function notation is a fundamental skill in mathematics and is crucial for success in higher-level mathematics courses and applications.

Our initial equation is $9x + 3y = 12$. To express this equation in function notation with x as the independent variable, we need to isolate y on one side of the equation. This process involves rearranging the equation to solve for y in terms of x. We begin by subtracting $9x$ from both sides of the equation, which gives us $3y = -9x + 12$. Next, we divide both sides of the equation by 3 to isolate y, resulting in $y = -3x + 4$. Now that we have y expressed in terms of x, we can write the function using function notation. We replace y with f(x), which represents the value of the function when the input is x. Therefore, the function in function notation is $f(x) = -3x + 4$. This notation clearly shows that the output of the function, f(x), is determined by the input x. The equation $f(x) = -3x + 4$ is a linear function, meaning that its graph is a straight line. The coefficient of x, which is -3, represents the slope of the line, and the constant term, 4, represents the y-intercept. This form of the equation is particularly useful for analyzing the behavior of the function, such as its rate of change and its points of intersection with the coordinate axes. Understanding how to convert equations into function notation is a crucial skill in mathematics, as it allows us to represent mathematical relationships in a clear and concise manner, facilitating further analysis and manipulation.

Therefore, the correct way to write the given equation $9x + 3y = 12$ in function notation with x as the independent variable is $f(x) = -3x + 4$. This corresponds to option B. This notation is the standard way of representing a function where x is the independent variable and f(x) represents the dependent variable, which is the output of the function for a given input x. The equation $f(x) = -3x + 4$ clearly defines a linear function with a slope of -3 and a y-intercept of 4. The slope indicates the rate of change of the function, while the y-intercept indicates the point where the line crosses the y-axis. This function notation allows for easy evaluation of the function for different values of x. For example, to find the value of the function when x is 2, we substitute x with 2 in the equation, which gives us $f(2) = -3(2) + 4 = -6 + 4 = -2$. This means that the output of the function is -2 when the input is 2. Understanding the correct function notation is crucial for accurately representing and interpreting mathematical relationships, particularly in the context of functions and their properties. The ability to translate between different forms of equations and function notation is a fundamental skill in mathematics, allowing for a deeper understanding of mathematical concepts and their applications.

Option A, $f(y) = -rac{1}{3}y + rac{4}{3}$, is incorrect because it expresses x as a function of y, which is the opposite of what the problem asks for. In this option, y is treated as the independent variable, and x is expressed as the dependent variable. This is a valid function, but it represents the inverse relationship of the original equation. To obtain this function, we would need to solve the original equation for x in terms of y. Starting with $9x + 3y = 12$, we first subtract $3y$ from both sides, which gives us $9x = -3y + 12$. Then, we divide both sides by 9 to isolate x, resulting in $x = -rac{1}{3}y + rac{4}{3}$. This equation can be written in function notation as $f(y) = -rac{1}{3}y + rac{4}{3}$, where f(y) represents the value of x for a given y. However, since the problem specifically asks for the function with x as the independent variable, this option is incorrect. It's important to carefully consider the question's requirements regarding the independent and dependent variables when expressing equations in function notation. The correct function notation should reflect the relationship where the output is expressed in terms of the specified independent variable. Understanding the difference between a function and its inverse is crucial for accurately interpreting and representing mathematical relationships.

In conclusion, converting the equation $9x + 3y = 12$ to function notation with x as the independent variable results in $f(x) = -3x + 4$. This process involves isolating y and then replacing it with f(x). Understanding function notation is a key skill in mathematics, enabling clear and concise representation of mathematical relationships. Function notation is a fundamental concept in mathematics that extends beyond simple linear equations. It is used extensively in calculus, algebra, and various other branches of mathematics to represent complex relationships and transformations. The ability to work with function notation is essential for understanding and applying mathematical principles in diverse contexts. Function notation not only simplifies the representation of mathematical functions but also provides a powerful tool for analyzing and manipulating these functions. By using function notation, mathematicians can easily describe transformations, compositions, and other operations on functions, making it a cornerstone of mathematical analysis. Furthermore, function notation plays a crucial role in mathematical modeling, where real-world phenomena are represented using mathematical functions. The clear and concise nature of function notation allows for the development of accurate and efficient models, which are essential in fields such as physics, engineering, and economics. Therefore, mastering function notation is not only important for academic success in mathematics but also for practical applications in various scientific and technical disciplines.