Converting Equations To Function Notation A Step-by-Step Guide

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Isolating y

To rewrite the given equation in function notation, the first crucial step involves isolating the dependent variable yy. In the original equation, yβˆ’6xβˆ’9=0y - 6x - 9 = 0, we need to manipulate the equation to get yy by itself on one side. This is achieved by performing algebraic operations that maintain the equality of the equation. To accomplish this, we can add 6x6x to both sides of the equation. This cancels out the βˆ’6x-6x term on the left side, leaving us with yβˆ’9=6xy - 9 = 6x. Then, we add 99 to both sides of the equation. This cancels out the βˆ’9-9 term on the left side, finally isolating yy. After these two steps, the equation transforms into y=6x+9y = 6x + 9. This form clearly expresses yy in terms of xx, which is a prerequisite for writing the equation in function notation. Isolating the dependent variable is a fundamental technique in algebra and is essential for solving equations and understanding relationships between variables. It allows us to express one variable explicitly in terms of others, making it easier to analyze and manipulate the equation. In this case, isolating yy allows us to see the direct relationship between xx and yy, which is crucial for understanding the function that the equation represents. The process of isolating a variable often involves applying inverse operations to both sides of the equation. For example, adding 6x6x is the inverse operation of subtracting 6x6x, and adding 99 is the inverse operation of subtracting 99. By applying these inverse operations, we can systematically eliminate terms from one side of the equation until only the desired variable remains. It's important to perform the same operation on both sides of the equation to maintain equality. This ensures that the resulting equation is equivalent to the original equation, meaning it has the same solutions. Isolating the dependent variable is not only necessary for writing equations in function notation but also for graphing equations, solving systems of equations, and performing other algebraic manipulations. It is a versatile technique that is used throughout mathematics and related fields.

Function Notation

Having successfully isolated yy, we can now express the equation in function notation. This is where we replace yy with f(x)f(x). The expression f(x)f(x) is read as "f of x" and represents the value of the function ff at the input xx. It's a standard notation used in mathematics to indicate that a variable is a function of another variable. In our case, since y=6x+9y = 6x + 9, we can rewrite this as f(x)=6x+9f(x) = 6x + 9. This equation explicitly states that the value of the function ff at xx is given by 6x+96x + 9. The function notation f(x)f(x) is not merely a symbolic replacement for yy; it carries significant meaning and provides a powerful way to represent and work with functions. It emphasizes the functional relationship between xx and yy, where for each input xx, there is a unique output f(x)f(x). This notation allows us to easily refer to the output of the function for a specific input. For example, to find the value of the function when x=2x = 2, we simply substitute 22 for xx in the expression f(x)=6x+9f(x) = 6x + 9, which gives us f(2)=6(2)+9=21f(2) = 6(2) + 9 = 21. Function notation also facilitates the composition of functions, where the output of one function becomes the input of another. This is a fundamental concept in mathematics and is used extensively in calculus and other advanced topics. The use of function notation also extends beyond simple algebraic functions. It is used to represent trigonometric functions, exponential functions, logarithmic functions, and many other types of functions. In each case, the notation f(x)f(x) represents the value of the function ff at the input xx, regardless of the specific nature of the function. Therefore, understanding function notation is crucial for developing a strong foundation in mathematics and for progressing to more advanced topics.

Identifying the Correct Answer

After rewriting the equation yβˆ’6xβˆ’9=0y - 6x - 9 = 0 in function notation as f(x)=6x+9f(x) = 6x + 9, we can now identify the correct answer among the given options. By carefully comparing our result with the options provided, we can see that option A, f(x)=6x+9f(x) = 6x + 9, perfectly matches our derived equation. Therefore, option A is the correct answer. The other options can be ruled out as they do not represent the same relationship between xx and yy as the original equation. Option B, f(x) = rac{1}{6}x + rac{3}{2}, represents a different linear function with a different slope and y-intercept. Option C, f(y)=6y+9f(y) = 6y + 9, is incorrect because it expresses xx as a function of yy, rather than yy as a function of xx, which is what the question asked for. In function notation, the variable inside the parentheses represents the independent variable, and the expression on the other side of the equation represents the dependent variable. Therefore, f(x)f(x) means that yy (or f(x)f(x)) is a function of xx, while f(y)f(y) would mean that xx is a function of yy. Identifying the correct answer often involves a process of elimination, where we systematically rule out incorrect options based on our understanding of the concepts involved. In this case, by understanding function notation and the steps involved in rewriting an equation in function notation, we can confidently identify the correct answer and eliminate the incorrect ones. This process of careful analysis and comparison is a valuable skill in mathematics and other problem-solving contexts. It allows us to approach problems systematically and to arrive at the correct solution by considering all the available information and applying the relevant concepts and techniques. Therefore, developing this skill is crucial for success in mathematics and beyond. The ability to identify the correct answer also demonstrates a thorough understanding of the underlying concepts and principles. In this case, it shows a clear understanding of function notation, the process of isolating variables, and the relationship between equations and functions.

Final Answer: The final answer is f(x)=6x+9\boxed{f(x)=6x+9}