Linear Function: Find F(x) = Kx + T & Its Root

Linear functions are a fundamental concept in mathematics, especially in algebra and calculus. These functions, which can be written in the form f(x) = kx + t, where k and t are constants, represent straight lines when graphed on a coordinate plane. Understanding linear functions is crucial because they model many real-world scenarios, from simple relationships like the cost of items (where x might be the quantity, k the price per item, and t a fixed cost) to more complex models in physics, economics, and engineering. Guys, let's dive deep into the properties of linear functions and how we can solve for their specific forms given certain conditions. The beauty of a linear function lies in its simplicity and predictability. The constant k determines the slope of the line, indicating how steeply it rises or falls. A positive k means the line slopes upwards from left to right, while a negative k indicates a downward slope. The constant t is the y-intercept, the point where the line crosses the y-axis. This is the value of f(x) when x is zero. These two constants, the slope and the y-intercept, completely define a unique straight line. When you're given two points on a line, you essentially have enough information to find these constants and, thus, the equation of the line. This is because each point provides an x and f(x) value that must satisfy the equation f(x) = kx + t. We can use these points to set up a system of equations and solve for k and t. So, when you see a problem involving points on the graph of a linear function, remember that you're being given clues to unlock the specific equation of that line. It's like detective work, where each point is a piece of evidence that helps you reveal the underlying mathematical relationship. The process of finding the equation of a line from points is not just a mathematical exercise; it's a way of translating real-world data into a mathematical model. This model can then be used to make predictions, analyze trends, and understand relationships between variables. For example, if you know the cost of a service for two different durations, you can find the linear function that describes the cost as a function of time. This allows you to predict the cost for any duration and make informed decisions. This practical applicability is what makes linear functions such an important tool in various fields.

Problem Statement: Finding the Function f(x)

In this problem, we're given that f(x) = kx + t is a linear function, and we know two points that lie on its graph: (-1, 3) and (0, -1). Our task is twofold: first, we need to find the specific values of k and t that define this function, and then, we need to find the root of the equation f(x) = 0. This problem is a classic example of how to work with linear functions, and it demonstrates how we can use given information to determine the function's equation. Guys, let's break this down step-by-step. The first part of the problem asks us to find the function f(x), which means we need to find the values of the constants k and t. We are given two points, each of which provides us with a relationship between x and f(x). The point (-1, 3) tells us that when x = -1, f(x) = 3. Similarly, the point (0, -1) tells us that when x = 0, f(x) = -1. We can use these two pieces of information to create two equations. By substituting these values into the general form of the linear function, f(x) = kx + t, we create a system of two linear equations in two variables, k and t. This system can then be solved using various methods, such as substitution or elimination. Solving this system will give us the unique values of k and t that define our function. Once we have these values, we can write the specific equation for f(x). The second part of the problem asks us to find the root of the equation f(x) = 0. The root of an equation is the value of x that makes the equation true. In the context of a linear function, the root is the x-intercept, the point where the line crosses the x-axis. To find the root, we simply set f(x) equal to zero and solve for x. This involves substituting the values of k and t that we found in the first part of the problem and then using algebraic manipulation to isolate x. Finding the root is an important skill because it tells us where the function's value changes sign. In many real-world applications, the root represents a critical point, such as a break-even point in business or a zero-crossing in signal processing. Guys, the ability to solve for these constants and the root of the equation is crucial for anyone working with linear functions.

Part a: Finding the Coefficients of f(x)

To find the coefficients k and t, we'll use the two points given: (-1, 3) and (0, -1). These points tell us that: When x = -1, f(x) = 3; and when x = 0, f(x) = -1. Let's substitute these values into the equation f(x) = kx + t. Guys, this is where the magic happens, we are turning points into equations! For the point (-1, 3), we have 3 = k(-1) + t, which simplifies to 3 = -k + t. For the point (0, -1), we have -1 = k(0) + t, which simplifies to -1 = t. Notice that the second equation directly gives us the value of t: t = -1. This is a common occurrence when one of the given points has an x-coordinate of 0, as it immediately reveals the y-intercept. Now that we know t = -1, we can substitute this value into the first equation to solve for k. Substituting t = -1 into 3 = -k + t gives us 3 = -k - 1. Adding 1 to both sides, we get 4 = -k. Multiplying both sides by -1, we find k = -4. So, we have found the values of both constants: k = -4 and t = -1. This means the specific equation for our linear function is f(x) = -4x - 1. This equation tells us that the line has a negative slope of -4, meaning it goes downwards from left to right, and it crosses the y-axis at -1. Guys, isn't it cool how two simple points can completely define a line? Finding these coefficients is like unlocking the code to the line's behavior. We now know exactly how this function will behave for any given value of x. We can plug in any x and instantly find the corresponding f(x). This is the power of having the equation of a function. This process of finding the coefficients is not just limited to linear functions. Similar techniques can be used to find the parameters of other types of functions, such as quadratic or exponential functions, given enough information. The key is to use the given data to create a system of equations and then solve for the unknowns. This is a fundamental skill in mathematical modeling and data analysis. Guys, mastering this technique opens the door to understanding and predicting a wide range of phenomena.

Part b: Finding the Root of the Equation f(x) = 0

Now that we have the function f(x) = -4x - 1, let's find the root of the equation f(x) = 0. The root is the value of x for which f(x) equals zero. In other words, it's the x-coordinate of the point where the line crosses the x-axis. Guys, think of the root as the function's secret zero point! To find the root, we set f(x) to 0 and solve for x: 0 = -4x - 1. We need to isolate x on one side of the equation. First, let's add 1 to both sides: 1 = -4x. Now, we'll divide both sides by -4: x = -1/4. So, the root of the equation f(x) = 0 is x = -1/4. This means the line represented by the function f(x) = -4x - 1 crosses the x-axis at the point (-1/4, 0). Understanding the root of a function is crucial for many applications. It tells us where the function's value changes sign. For a linear function, the root is the only point where this sign change occurs. If we visualize the graph of the function, we can see that the line is negative for x < -1/4 and positive for x > -1/4. Guys, this knowledge can be incredibly valuable! In practical terms, the root can represent a break-even point, a threshold, or a critical value in a system. For example, in a business context, the root of a cost-profit function might represent the point at which the business starts making a profit. In a scientific context, the root might represent a point of equilibrium or a change in state. The process of finding the root of a linear equation is relatively straightforward, as we've seen. However, the concept of a root extends to more complex functions, such as quadratic, polynomial, and trigonometric functions. For these functions, finding the roots can be more challenging, and it may involve using techniques such as factoring, the quadratic formula, or numerical methods. Guys, the fundamental idea remains the same: we're looking for the values of x that make the function equal to zero. This is a powerful concept that underlies much of mathematical analysis and problem-solving. So, whether you're working with a simple linear function or a more complex equation, remember the significance of the root – it's a key piece of information about the function's behavior.

Conclusion

In summary, we've successfully determined the linear function f(x) = -4x - 1 given two points on its graph, and we've found its root to be x = -1/4. This exercise highlights the fundamental process of working with linear functions: using given information to solve for unknown coefficients and finding critical points like the root. Guys, this is like solving a puzzle where each piece of information fits together to reveal the complete picture! Understanding these concepts is crucial for anyone studying mathematics or related fields. Linear functions are the building blocks of many more advanced mathematical models, and the techniques we've used here can be applied to a wide range of problems. The ability to find the equation of a line given points, and to find the roots of equations, is a fundamental skill that will serve you well in many contexts. From modeling real-world phenomena to solving abstract mathematical problems, linear functions are a powerful tool in your mathematical arsenal. So, keep practicing, keep exploring, and keep building your understanding of these essential concepts. Guys, the world of mathematics is full of exciting challenges and rewarding discoveries, and linear functions are just the beginning!