Commission Function Problem Sales, Commission, And Domain

Problema: Un vendedor gana una comisión de $50 más el 5% de las ventas realizadas. Encuentra la función que relaciona la comisión total (f(x)) con el valor de las ventas (x). Determina el dominio de la función y completa la tabla de valores para x = 0.

Hey guys! Let's dive into this math problem about a salesperson's commission. It's a pretty common scenario, and understanding how to set up the function and figure out the domain is super useful. We're going to break it down step by step so it's crystal clear. First, we need to identify the function that connects the total commission, which we'll call f(x), with the value of the sales, represented by x. The salesperson gets a flat $50, which is like their base pay, and then they get an extra 5% of whatever they sell. So, how do we put that into math terms? The key here is understanding percentages and how they translate into decimals. Remember, 5% is the same as 0.05. So, for every dollar the salesperson sells, they get an extra $0.05. If they sell $100, they get 5% of that, which is $5. The variable x represents the total sales amount. To calculate the commission earned from sales alone, we multiply the sales amount x by 0.05. This gives us 0.05x, which is the amount earned from the 5% commission on sales. Now, we can't forget the base pay of $50. This is a fixed amount, meaning it doesn't change no matter how much or how little the salesperson sells. To get the total commission, we need to add this base pay to the commission earned from sales. So, the total commission f(x) is the base pay of $50 plus 0.05x. Putting it all together, we get the function f(x) = 50 + 0.05x. This is the function that links the total commission f(x) to the value of sales x. This equation tells us exactly how to calculate the salesperson's total earnings based on their sales performance. The $50 is a constant, representing the fixed amount they earn regardless of sales, and the 0.05x represents the variable part of their earnings, which directly depends on their sales volume. This function is a linear function, meaning it graphs as a straight line. The 0.05 is the slope of the line, indicating how much the commission increases for every additional dollar in sales. The 50 is the y-intercept, which is the commission when sales are zero. Understanding linear functions like this is crucial in many real-world applications, from calculating paychecks to predicting costs and revenue in business. So, this equation, f(x) = 50 + 0.05x, is the bread and butter of this problem. It encapsulates the relationship between sales and total commission in a neat, mathematical package. Now that we have our function, we need to think about what it really means and what values we can plug into it. That's where the domain comes in.

Domain of the Function

Now, let's talk about the domain of our commission function. The domain, guys, is all about figuring out what values of x (the sales amount) actually make sense in this situation. It's not just about what you can plug into the equation, but also what makes logical sense in the real world. In the context of sales, the value of sales (x) can't be negative. You can't sell a negative amount of stuff, right? So, x has to be greater than or equal to zero. Makes sense so far? Now, is there an upper limit to the value of sales? Well, theoretically, a salesperson could sell an unlimited amount, so there's no specific maximum value for x. However, in practical terms, there might be some realistic upper limit depending on the context of the job. For example, if the salesperson is selling houses, there's a limit to how many houses they can sell in a given period. But for the sake of this mathematical model, we'll assume there's no upper limit. So, the domain of the function is all non-negative real numbers. We can write this in a few different ways. We could say x ≥ 0. Or, we can use interval notation, which looks like this: [0, ∞). The square bracket on the 0 means that 0 is included in the domain, and the infinity symbol (∞) means that the domain extends infinitely in the positive direction. It's super important to think about the domain when you're working with functions because it tells you the range of inputs that are valid and meaningful. If you plug in a value that's outside the domain, you might get a mathematical result, but it won't make any sense in the real-world scenario. For example, if we plugged in x = -100 into our function, we'd get f(-100) = 50 + 0.05(-100) = 50 - 5 = $45. While this is a valid mathematical calculation, it doesn't make sense in the context of sales. You can't sell negative $100 worth of goods. So, understanding the domain helps us make sure that our results are not only mathematically correct but also logically sound. Now that we've nailed down the function and its domain, let's move on to filling in the table of values. This will give us a few specific points that we can use to visualize the function and understand how the commission changes as sales increase. Remember, the domain restricts the possible values for the input (x) to make the output (f(x)) meaningful in a real-world context. It's all about making sure our mathematical model reflects reality.

Table of Values for x = 0

Alright, let's complete this table of values! We're asked to find the commission when x = 0. This is actually a pretty straightforward calculation, and it will give us a key point to understand our function. Remember our commission function: f(x) = 50 + 0.05x. This function tells us exactly how to calculate the total commission (f(x)) based on the value of sales (x). When x = 0, it means the salesperson made no sales at all. But even if they don't sell anything, they still get their base commission, right? To find the commission when x = 0, we just plug 0 into our function: f(0) = 50 + 0.05(0). Now, anything multiplied by 0 is 0, so 0.05(0) = 0. This simplifies our equation to: f(0) = 50 + 0. So, f(0) = 50. This means that when the salesperson's sales are $0, their total commission is $50. This makes perfect sense because the $50 is the fixed base commission they receive regardless of sales. In the context of a graph, this point (0, 50) is the y-intercept of our linear function. It's the point where the line crosses the y-axis, and it represents the starting point of the commission – the amount earned even before any sales are made. Filling in this value in a table is super helpful because it gives us a concrete example of how the function works. It's also the first step in understanding how the commission changes as sales increase. We could now fill in other values for x to see how the commission goes up as the salesperson sells more. For example, if we wanted to know the commission when x = $100, we'd plug that into our function: f(100) = 50 + 0.05(100) = 50 + 5 = $55. So, if the salesperson sells $100 worth of stuff, their commission would be $55. We could do this for several different values of x and then plot those points on a graph to visualize the relationship between sales and commission. This is a great way to really understand what the function is telling us. In summary, finding the commission when x = 0 is a simple but crucial step in understanding the commission function. It tells us the base commission and gives us a starting point for analyzing how the commission changes with sales. Plus, it highlights the importance of the constant term in the function, which represents the fixed part of the commission.

In this whole process, guys, we've not only solved a math problem, but we've also explored a real-world scenario. We've seen how math can be used to model things like sales commissions, and how understanding functions and their domains can help us make sense of the world around us. Now you have the tools to calculate the commission for any sales amount, and you understand why the domain is important. Keep practicing, and you'll become a pro at these types of problems!