Solving F(x) = 3x - 2 Calculating Function Values And Inverse

Hey guys! Today, we're diving deep into the world of functions, specifically the function f(x) = 3x − 2. We'll break down how to calculate its values for different inputs and, more excitingly, how to find its inverse. So, buckle up and let's get started!

Understanding the Function f(x) = 3x − 2

Let's first understand our main function, f(x) = 3x − 2. This is a linear function, which means its graph will be a straight line. The function takes an input x, multiplies it by 3, and then subtracts 2. Think of it as a little machine: you put a number in, and it spits out another number according to this rule. Understanding this basic function is key to tackling the problems that follow. Linear functions are fundamental in mathematics, appearing in various contexts, from simple algebra to more advanced calculus. They are characterized by their constant rate of change, which in this case is 3, the coefficient of x. This means that for every increase of 1 in x, the value of f(x) increases by 3. The constant term, -2, represents the y-intercept of the line, the point where the line crosses the y-axis. Visualizing this function as a line can provide valuable insights into its behavior. For instance, we can easily see that as x increases, f(x) also increases, and vice versa. This understanding of the function's behavior will be particularly useful when we discuss its inverse. The inverse function, as we'll see, essentially reverses this process, taking the output of f(x) as its input and returning the original x value. Mastering the concept of linear functions like f(x) = 3x − 2 provides a solid foundation for understanding more complex functions and mathematical concepts. So, let's keep this fundamental understanding in mind as we move forward and tackle the specific calculations and the inverse function.

Calculating Function Values

a) Finding f(0)

So, the first task is to find f(0). What does this mean? It simply means we need to substitute x with 0 in our function, f(x) = 3x − 2. Let's do it! When we replace x with 0, we get f(0) = 3(0) − 2. Now, this is straightforward arithmetic. 3 multiplied by 0 is 0, so we have f(0) = 0 − 2. Therefore, f(0) = -2. Easy peasy, right? This tells us that when the input is 0, the output of our function is -2. Graphically, this corresponds to the point (0, -2) on the line representing the function. It's a crucial point, as it's the y-intercept, where the line crosses the vertical axis. Understanding how to calculate function values like f(0) is fundamental in mathematics. It allows us to determine the behavior of the function at specific points and to create a visual representation of the function's graph. In the case of a linear function, knowing the y-intercept and the slope (which is 3 in our case) is enough to sketch the entire line. The process of substituting values into a function is a recurring theme in mathematics. Whether we're dealing with simple linear functions or complex trigonometric or exponential functions, the underlying principle remains the same: replace the variable with the given value and simplify. This simple yet powerful technique allows us to explore the vast landscape of mathematical functions and their applications. So, remember this basic step, and you'll be well-equipped to tackle more challenging problems.

b) Finding f(4)

Next up, we need to calculate f(4). This is the same process as before, just with a different input value. We're going to substitute x with 4 in the function f(x) = 3x − 2. So, we get f(4) = 3(4) − 2. Now, let's do the math. 3 multiplied by 4 is 12, so we have f(4) = 12 − 2. Subtracting 2 from 12 gives us 10. Therefore, f(4) = 10. Great job! This means that when the input is 4, the output of our function is 10. On the graph of the function, this corresponds to the point (4, 10). Plotting this point along with the y-intercept (0, -2) that we found earlier helps us to visualize the line representing the function. These two points are sufficient to draw the entire line, illustrating the linear nature of the function. Calculating function values for different inputs is essential for understanding the behavior of the function over its entire domain. By evaluating the function at various points, we can get a sense of how the output changes as the input changes. This is particularly important in applications where we need to model real-world phenomena using mathematical functions. For instance, if our function represented the cost of producing a certain number of items, then f(4) = 10 would tell us that it costs 10 units of currency to produce 4 items. The ability to calculate function values is a fundamental skill in mathematics and is a building block for more advanced concepts such as calculus and differential equations. So, practice these calculations, and you'll be well on your way to mastering the world of functions!

Finding the Inverse Function

c) Determining the Inverse Function f −1 (x)

Okay, now for the exciting part: finding the inverse function, f −1(x). The inverse function essentially