Unveiling The Secrets Of A Logistic Function: A Comprehensive Guide

Hey math enthusiasts! Let's dive into the fascinating world of functions, specifically a logistic function. We'll break down the function step-by-step, making sure everyone understands, regardless of your math background. We'll start by graphing it, calculating some values, and then explore its behavior. Get ready to have some fun, guys!

Graphing the Logistic Function: A Visual Journey

Alright, first things first, let's graph the function f(x) = rac{198}{1 + 3e^{-2x}} from x = 0 to x = 10. To do this, we can either use a graphing calculator (like a TI-84 or Desmos – which is super user-friendly and online!) or plot points manually. I highly recommend using a graphing calculator, it's so much easier and faster. Essentially, we are going to be plotting the function in the interval of x = 0 to x = 10. What happens here is, as the value of 'x' increases, the value of $e^{-2x}$ decreases. This is because the exponent is negative, which means we are essentially dividing by an increasing number. For instance, when x = 0, $e^{-2x}$ is $e^0 = 1$. As x becomes larger (like, say x = 5), $e^{-2x}$ becomes $e^{-10}$, which is a very small number (approximately 0.000045). Consequently, the denominator (1 + 3e-2x) gets closer and closer to 1, as the $3e^{-2x}$ part approaches 0. The graph starts relatively low, then curves upward, and gradually levels off, approaching a horizontal line. This characteristic 'S-shape' is what defines a logistic function. The range of the function is all the possible values of f(x) on the graph. The domain is the possible values of x. In this case, the domain is [0,10]. The range is going to be a number between 0 and 198. The graph would increase, as the value of x increase, the value of f(x) would also increase. This would be a continuous function, no breaks, or discontinuities. So, start by inputting the function into your graphing tool. Then, set the x-axis to range from 0 to 10. The y-axis will adjust automatically, and the graph should look like a stretched-out S-curve. The y-axis, the maximum value that it can have is 198. Don't worry if it's not perfect. The general shape is the most important thing. You will see that the graph starts increasing rapidly, then the rate of increase slows down, and it eventually levels off.

So, grab your graphing tool, plot the function, and take a look at the beautiful S-curve. It's really that simple! Let's explore some key characteristics, so we can get a better understanding of the concept.

Finding Specific Values: Decoding f(0) and f(10)

Now, let's find the specific values of our function at two points: x = 0 and x = 10. This involves plugging those x-values into our equation and solving for f(x). This will help us understand the behavior of the function at these particular points. It's like taking snapshots of the graph at specific locations. Let's calculate $f(0)$ and $f(10)$ step by step.

First, for f(0), we substitute x = 0 into the equation: f(0) = rac{198}{1 + 3e^{-2(0)}}. Remember that anything raised to the power of 0 is 1. Thus, $e^{-2(0)} = e^0 = 1$. This simplifies to: f(0) = rac{198}{1 + 3(1)} = rac{198}{4} = 49.5. So, when x = 0, the value of the function is 49.5.

Next, let's find f(10). We substitute x = 10 into the equation: f(10) = rac{198}{1 + 3e^{-2(10)}} = rac{198}{1 + 3e^{-20}}. Now, $e^{-20}$ is an extremely small number, almost equal to zero (approximately 2.06 x 10^-9). So, $3e^{-20}$ is also very close to zero. The equation becomes f(10) = rac{198}{1 + 0} = rac{198}{1} = 198. Therefore, as x approaches 10, the function's value gets very close to 198. These calculations give us concrete data points on the graph. By finding f(0) and f(10), we've identified specific points on the curve and confirmed the general trend: the function starts at a certain value and then increases as x increases. This gives us a clearer understanding of how the logistic function behaves as x changes. We've essentially pinpointed the function's behavior at two critical locations.

Increasing or Decreasing? Unveiling the Function's Trend

Now, let's address a key question: Is our logistic function increasing or decreasing? This means, as the value of x increases, does the value of f(x) increase or decrease? Looking at the graph, or analyzing the function's behavior, is really easy to tell. As we move from x = 0 to x = 10, the function's value increases. This is a very important concept. Another way to think about it is as we said before, $f(0) = 49.5$, and $f(10)$ is very close to 198. Since f(10) is greater than f(0), this confirms that the function is indeed increasing over the given interval. To confirm this mathematically, we can examine the derivative of the function (though that's a bit more advanced). The derivative of a function tells us the rate of change. If the derivative is positive, then the function is increasing; if the derivative is negative, the function is decreasing. The derivative of this logistic function is always positive. The function is going up as x is increasing. So, for all values between 0 and 10, the function will be increasing. This property is characteristic of logistic functions in general. Logistic functions always increase at first, then level off, approaching a horizontal asymptote. It starts with a small slope, then becomes steeper, then eventually flattens out. The increasing nature of the function is a direct consequence of the equation and its components. The term e^-2x ensures that as x increases, the denominator of the function also increases, resulting in the function values increasing. So, in our case, the function is always increasing from x = 0 to x = 10. It's a key attribute of this type of function!

The Limiting Value: What Happens in the Long Run?

Finally, let's explore the