Expressing The Relationship Between S And T In Function Notation

by Viktoria Ivanova 65 views

Hey guys! Today, we're diving into the fascinating world of functions and variables, specifically exploring the relationship between s and t when t depends on s, and their connection is defined by the equation 2t + 4s = 3 - 6s. Our mission? To express this relationship using the correct function notation. Let's break it down step by step, making sure everyone's on board.

Understanding Dependent and Independent Variables

Before we jump into the equation, let's quickly recap what it means for one variable to depend on another. In our case, t is the dependent variable, meaning its value hinges on the value of s, which is the independent variable. Think of it like this: s is the input, and t is the output. We're trying to find a way to write t as a function of s, denoted as t(s). In this first step, we will focus on identifying the dependent and independent variables. In the equation 2t + 4s = 3 - 6s, t is expressed in terms of s. This means that the value of t is determined by the value of s. Therefore, t is the dependent variable, and s is the independent variable. This distinction is crucial for understanding how to represent the relationship in function notation. The dependent variable is the one whose value changes in response to changes in the independent variable. Conversely, the independent variable is the one that can be changed freely, and its value affects the dependent variable. When we are asked to express a relationship in function notation, we are essentially being asked to isolate the dependent variable and express it as a function of the independent variable. The function notation, in this case, will take the form t(s) = [expression], where the expression on the right side involves only the independent variable s. This notation clearly shows that the value of t is a function of s, and it allows us to easily evaluate t for different values of s. Understanding this fundamental concept of dependent and independent variables is the first step towards correctly representing the relationship between s and t in function notation. It sets the stage for the subsequent steps, where we will manipulate the given equation to isolate t and express it as a function of s.

Isolating t: Our First Big Step

Our main goal here is to rewrite the equation so that t is all by itself on one side. This is what it means to isolate the dependent variable. Currently, our equation looks like this: 2t + 4s = 3 - 6s. The first step in isolating t is to get rid of the term 4s on the left side. We can do this by subtracting 4s from both sides of the equation. This maintains the balance of the equation and moves us closer to our goal. Subtracting 4s from both sides, we get: 2t + 4s - 4s = 3 - 6s - 4s. Simplifying this gives us: 2t = 3 - 10s. Now we have 2t isolated on the left side, but we still need to get t by itself. To do this, we divide both sides of the equation by 2. This will undo the multiplication by 2 on the left side and leave us with t alone. Dividing both sides by 2, we get: (2t) / 2 = (3 - 10s) / 2. Simplifying this gives us: t = (3 - 10s) / 2. Now we have t expressed in terms of s, which is exactly what we needed to do. We have successfully isolated the dependent variable t. This is a crucial step because it allows us to express the relationship between s and t in function notation. The next step is to rewrite the expression for t in the standard function notation, which will clearly show the functional relationship between the two variables. This isolation process is a common technique in algebra and is essential for solving equations and expressing relationships between variables. By carefully following these steps, we have transformed the original equation into a form that explicitly shows how t depends on s.

Expressing t in Function Notation

Now that we have t = (3 - 10s) / 2, let's translate this into function notation. Remember, function notation is a fancy way of saying t is a function of s, written as t(s). This is where things get really cool! We simply replace t with t(s) on the left side of the equation. So, t becomes t(s), and everything else stays the same. This gives us: t(s) = (3 - 10s) / 2. And there you have it! We've successfully expressed the relationship between s and t in function notation. This notation tells us exactly how to find the value of t for any given value of s. For example, if we wanted to find the value of t when s = 1, we would substitute 1 for s in the function: t(1) = (3 - 10(1)) / 2 = (3 - 10) / 2 = -7 / 2. So, when s is 1, t is -7/2. Function notation is a powerful tool because it allows us to easily see the relationship between variables and to evaluate the function for different inputs. It also makes it clear which variable is the independent variable and which is the dependent variable. In this case, s is the input (the independent variable), and t(s) is the output (the dependent variable). By expressing the relationship in function notation, we have made it clear, concise, and easy to use. This is the final step in our process, and it completes our task of representing the relationship between s and t in the correct functional form.

Simplifying the Function (Optional)

While t(s) = (3 - 10s) / 2 is perfectly correct, we can make it look even cleaner by simplifying the expression a bit. This is totally optional, but it sometimes makes the function easier to work with. We can split the fraction into two parts: t(s) = 3/2 - (10s) / 2. Then, we can simplify the second term: t(s) = 3/2 - 5s. Both t(s) = (3 - 10s) / 2 and t(s) = 3/2 - 5s represent the same relationship, just in slightly different forms. The simplified form can sometimes be more convenient for certain calculations or analyses. For instance, it's easier to see the slope and y-intercept of the function in the simplified form. The slope is the coefficient of s, which is -5, and the y-intercept is the constant term, which is 3/2. These values are not as immediately apparent in the original form of the function. Simplifying functions is a common practice in mathematics, and it often leads to a more intuitive understanding of the function's properties. However, it's important to remember that the simplified form is equivalent to the original form, and both forms are equally valid. In this case, simplifying the function gives us t(s) = 3/2 - 5s, which is a linear function with a slope of -5 and a y-intercept of 3/2. This simplified form provides additional insights into the relationship between s and t. Whether you choose to simplify the function or not, the key is to understand the underlying relationship and be able to express it in function notation.

Key Takeaways

So, what have we learned today, guys? We've tackled how to take an equation relating two variables (s and t), where one depends on the other, and translate it into function notation. Remember these crucial steps:

  1. Identify the dependent and independent variables.
  2. Isolate the dependent variable (t in this case).
  3. Express the relationship using t(s) notation.
  4. Simplify the function (optional, but often helpful).

By following these steps, you can confidently express relationships between variables in function notation, a fundamental skill in mathematics and beyond. The ability to correctly represent relationships between variables in function notation is essential for understanding and working with mathematical models. It allows us to clearly define how one variable changes in response to changes in another variable. Moreover, function notation provides a standardized way to communicate these relationships, making it easier for others to understand and use the results. In addition to the specific steps we've discussed, it's also important to develop a strong conceptual understanding of functions and variables. This includes understanding the difference between dependent and independent variables, the meaning of function notation, and the various ways in which functions can be represented (e.g., equations, graphs, tables). With a solid foundation in these concepts, you'll be well-equipped to tackle more advanced mathematical problems and applications. Keep practicing these skills, and you'll become a function notation pro in no time! Remember, mathematics is not just about memorizing formulas and procedures; it's about understanding the underlying concepts and developing the ability to apply them in different contexts.

By mastering function notation, you're not just learning a mathematical tool; you're developing a powerful way of thinking about relationships and dependencies, which is a valuable skill in many areas of life. Keep exploring, keep questioning, and keep learning! You've got this!