Function Notation Convert 6q = 3s - 9 With Q As Independent Variable

Jul 16, 2025 by ADMIN 69 views

Consider the Function Represented by the Equation 6q = 3s - 9 Write the Equation in Function Notation, Where q is the Independent Variable

In mathematics, function notation is a way to represent functions using a symbolic representation. It allows us to clearly express the relationship between an input and its corresponding output. When working with functions, it's crucial to identify the independent and dependent variables. The independent variable is the input to the function, while the dependent variable is the output, which depends on the input. In the given equation, 6q = 3s - 9, we are asked to write the equation in function notation where q is the independent variable. This means we need to express s as a function of q, denoted as f(q).

The core concept here is to isolate the dependent variable, s, on one side of the equation. By doing this, we can define s explicitly in terms of q, thus expressing it as a function of q. The process involves algebraic manipulation to rearrange the equation, ensuring that s is the subject of the formula. This transformation is critical because it allows us to directly determine the value of s for any given value of q. Once we have s isolated, we can replace it with the function notation f(q), which provides a clear and concise representation of the functional relationship. Function notation not only simplifies the representation of functions but also makes it easier to perform operations and transformations on them. It is a fundamental tool in algebra and calculus, enabling us to analyze and manipulate functions effectively. In the context of real-world applications, understanding function notation is essential for modeling relationships between different quantities, such as the relationship between time and distance, cost and quantity, or input and output in a system. Therefore, mastering the concept of function notation and the process of converting equations into this form is a crucial skill for anyone studying mathematics and its applications.

To express the equation 6q = 3s - 9 in function notation with q as the independent variable, we need to isolate s on one side of the equation. This involves a series of algebraic steps aimed at rearranging the terms so that s is expressed in terms of q. Let's break down the process step by step. First, we have the equation 6q = 3s - 9. Our goal is to get s by itself. The initial step is to add 9 to both sides of the equation. This maintains the balance of the equation and moves the constant term to the left side, resulting in 6q + 9 = 3s. Next, we need to isolate s further by getting rid of the coefficient 3. To do this, we divide both sides of the equation by 3. This step is crucial because it directly solves for s in terms of q. Dividing both sides by 3 gives us (6q + 9) / 3 = s. Now, we can simplify the left side of the equation. We divide each term in the numerator by 3: (6q / 3) + (9 / 3) = s. This simplifies to 2q + 3 = s. We have now successfully isolated s and expressed it in terms of q. The final step is to rewrite the equation using function notation. Since s is now a function of q, we can replace s with f(q). This gives us the function notation f(q) = 2q + 3. This notation clearly indicates that the value of the function, f, depends on the value of the input variable, q. The transformation process highlights the importance of algebraic manipulation in expressing relationships between variables. By isolating the dependent variable, we gain a clear understanding of how it changes in response to changes in the independent variable. This is a fundamental skill in mathematics and is essential for solving a wide range of problems in various fields.

Now that we have transformed the equation 6q = 3s - 9 into function notation, we have f(q) = 2q + 3. This representation clearly shows the relationship between q and s, where q is the independent variable and s is the dependent variable, expressed as a function of q. Let's examine the given options to identify the correct one. The options provided are:

f(q) = (1/2)q - (3/2)
f(q) = 2s + 3
f(s) = (1/2)s - (3/2)
f(q) = 2q + 3

Comparing our derived function, f(q) = 2q + 3, with the options, we can see that the fourth option, f(q) = 2q + 3, exactly matches our result. This confirms that this is the correct function notation for the given equation with q as the independent variable. The other options are incorrect because they either do not represent the correct relationship between q and s or they express the function in terms of the wrong independent variable. For instance, the first option, f(q) = (1/2)q - (3/2), has a different slope and y-intercept than our derived function. The second option, f(q) = 2s + 3, incorrectly expresses the function in terms of s instead of q. The third option, f(s) = (1/2)s - (3/2), correctly identifies s as the independent variable but does not match the equation we derived. Therefore, the process of transforming the equation and comparing it with the given options allows us to confidently identify the correct function notation. This underscores the importance of carefully performing each step in the algebraic manipulation and verifying the result against the available choices. The ability to accurately express equations in function notation is a valuable skill in mathematics, enabling us to clearly communicate relationships between variables and solve problems effectively. In summary, the correct function notation for the given equation 6q = 3s - 9, where q is the independent variable, is f(q) = 2q + 3.

In conclusion, to express the equation 6q = 3s - 9 in function notation where q is the independent variable, we follow a series of algebraic steps to isolate s and then rewrite the equation using function notation. The process involves adding 9 to both sides of the equation, dividing both sides by 3, and simplifying to obtain s = 2q + 3. Replacing s with f(q) gives us the function notation f(q) = 2q + 3. This notation clearly represents s as a function of q. Among the given options, f(q) = 2q + 3 is the correct answer. This exercise demonstrates the importance of understanding function notation and the algebraic techniques required to transform equations into this form. Function notation is a fundamental concept in mathematics, providing a concise and clear way to express relationships between variables. It is widely used in various branches of mathematics and its applications in science, engineering, and economics. The ability to correctly identify and express functions in this notation is crucial for solving problems and understanding mathematical concepts. Moreover, the process of algebraic manipulation, which involves rearranging equations to isolate variables, is a core skill in mathematics. It requires a solid understanding of algebraic principles and the ability to apply them accurately. By mastering these skills, students can confidently tackle more complex mathematical problems and gain a deeper appreciation for the power and elegance of mathematical notation. In summary, the function notation f(q) = 2q + 3 correctly represents the given equation 6q = 3s - 9 with q as the independent variable, highlighting the significance of function notation and algebraic manipulation in mathematical problem-solving.