Understanding Exponential Functions: A Closer Look At F(x) = 2e^x

Welcome to our deep dive into the fascinating world of exponential functions! Today, we're going to explore a specific example, $f(x) = 2e^x$, and calculate its values at various points. This function is a fantastic way to understand how exponential growth works and how the base of the exponent, in this case, the special number $e$, influences the output. We'll be calculating $f(-3)$, $f(-1)$, $f(0)$, $f(1)$, and $f(3)$, which will give us a great perspective on the function's behavior across negative, zero, and positive inputs. Understanding these calculations is crucial for many areas in mathematics, science, and finance, where exponential functions model phenomena like population growth, radioactive decay, and compound interest. So, let's get our calculators ready and embark on this mathematical journey to truly grasp the essence of $f(x) = 2e^x$.

Calculating $f(-3)$: Unpacking Negative Exponents

Let's start by calculating the function value at $x = -3$ for our function $f(x) = 2e^x$. When we substitute $-3$ for $x$, we get $f(-3) = 2e^{-3}$. Now, what does a negative exponent mean? It means we take the reciprocal of the base raised to the positive version of that exponent. So, $e^{-3}$ is the same as $\frac{1}{e^3}$. This tells us that for negative inputs, the function's value will be smaller than the coefficient $2$. We're essentially looking at $f(-3) = 2 \times \frac{1}{e^3}$. The number $e$, often called Euler's number, is an irrational number approximately equal to $2.71828$. So, $e^3$ is $e \times e \times e$, which is roughly $2.71828 \times 2.71828 \times 2.71828$, a value greater than $20$. Therefore, $\frac{1}{e^3}$ will be a small positive number. Multiplying this small positive number by $2$ will give us an even smaller positive number. This behavior is characteristic of exponential functions when dealing with negative inputs: the values approach zero as the input becomes more negative. It's important to remember that even though the exponent is negative, the result of $e$ raised to any real power, when multiplied by a positive coefficient like $2$, will always yield a positive result. This is because $e$ itself is a positive number, and raising a positive number to any real power results in another positive number. So, $f(-3)$ will be a small, positive value, illustrating the function's decay towards zero on the left side of the y-axis. This concept of decay is fundamental in understanding half-life calculations in physics and the diminishing returns in certain economic models. The precise value of $f(-3)$ is $2 \times e^{-3} \approx 2 \times 0.049787 = 0.099574$. This demonstrates that as $x$ becomes more negative, $f(x)$ gets progressively closer to $0$, without ever actually reaching it, showcasing the concept of a horizontal asymptote at $y=0$.

Evaluating $f(-1)$: A Step Closer to the Y-axis

Next, let's calculate $f(-1)$ for $f(x) = 2e^x$. Substituting $-1$ for $x$, we get $f(-1) = 2e^{-1}$. Similar to our previous calculation, $e^{-1}$ is equivalent to $\frac{1}{e}$. So, $f(-1) = 2 \times \frac{1}{e}$. Since $e \approx 2.71828$, $\frac{1}{e}$ is approximately $0.36788$. Therefore, $f(-1) \approx 2 \times 0.36788 = 0.73576$. Notice that this value is larger than $f(-3)$. This increase in value as we move from $x = -3$ to $x = -1$ illustrates the increasing nature of the exponential function, even in the negative domain. While the function is still less than the coefficient $2$, it's moving away from zero and towards the y-axis. The closer the negative exponent gets to zero, the larger the function's value becomes. This is because $e^0 = 1$, and as the exponent approaches $0$ from the negative side, $e^x$ approaches $1$. So, $f(x)$ approaches $2 \times 1 = 2$. This step-by-step evaluation helps us visualize the function's trajectory. It starts very close to zero for large negative $x$ values and gradually rises as $x$ increases towards $0$. This smooth upward curve in the negative x-axis region is a key characteristic of exponential growth functions. The calculation $f(-1) = 2e^{-1}$ is a practical example of how these functions model processes where an initial quantity is reduced over time but never completely eliminated, such as the diminishing charge in a discharging capacitor or the residual amount of a drug in a patient's system after a certain period.

Determining $f(0)$: The Crucial Y-intercept

Now, let's calculate the function value at $x = 0$, which is $f(0) = 2e^0$. A fundamental rule in exponents is that any non-zero number raised to the power of $0$ equals $1$. Therefore, $e^0 = 1$. Plugging this into our function, we get $f(0) = 2 \times 1 = 2$. This is a significant point for any exponential function of the form $f(x) = a imes b^x$. The value at $x=0$ always equals the coefficient $a$, because $b^0 = 1$. In our case, the coefficient is $2$. This means the point $(0, 2)$ is the y-intercept of the graph of $f(x) = 2e^x$. It's the point where the curve crosses the y-axis. This value is higher than both $f(-3)$ and $f(-1)$, as expected. The y-intercept is often a critical starting point or reference value in models. For instance, in population growth, it might represent the initial population size. In finance, it could be the initial investment amount before any interest is compounded. The fact that $f(0) = 2$ anchors the function's behavior. All negative inputs result in values less than $2$ (but greater than $0$), and all positive inputs will result in values greater than $2$, demonstrating the function's growth.

Computing $f(1)$: Stepping into Positive Growth

Moving on to positive inputs, let's calculate $f(1)$ for $f(x) = 2e^x$. Substituting $1$ for $x$, we get $f(1) = 2e^1$, which is simply $2e$. Since $e \approx 2.71828$, we have $f(1) \approx 2 \times 2.71828 = 5.43656$. Compare this value to $f(0) = 2$. We can see that $f(1)$ is significantly larger than $f(0)$. This jump illustrates the accelerating nature of exponential growth. For every unit increase in $x$, the function's value doesn't just increase by a fixed amount; it increases by a factor related to the base $e$. In this case, the value roughly multiplied by $e$ (approximately $2.71828$) as we moved from $x=0$ to $x=1$. This is the core concept of exponential growth: the rate of growth is proportional to the current value. This principle is observed in bacterial cultures doubling their population, investments growing with compound interest, and the spread of viruses. The value $f(1) = 2e$ represents a point on the curve where the function is clearly in its growth phase, rising steeply as $x$ increases. It's crucial to recognize that this growth is multiplicative, not additive. This is what distinguishes exponential functions from linear functions.

Calculating $f(3)$: Experiencing Rapid Growth

Finally, let's calculate $f(3)$ for $f(x) = 2e^x$. Substituting $3$ for $x$, we get $f(3) = 2e^3$. We already touched upon $e^3$ when we discussed $f(-3)$. $e^3$ is approximately $2.71828$ multiplied by itself three times, resulting in a value around $20.0855$. Therefore, $f(3) \approx 2 \times 20.0855 = 40.171$. Look at how dramatically this value has increased compared to $f(1)$ and even $f(0)$. This illustrates the power of exponential growth. As the input $x$ increases, the output $f(x)$ increases at an ever-increasing rate. The difference between $f(1)$ and $f(3)$ is substantial ($40.171 - 5.43656 = 34.73444$), and the difference between $f(0)$ and $f(1)$ was smaller ($5.43656 - 2 = 3.43656$). This widening gap is the hallmark of exponential functions in their growth phase. The value $f(3) = 2e^3$ shows that for positive inputs, especially those further away from zero, the function's value can become very large, very quickly. This rapid acceleration is why exponential growth, if unchecked, can lead to significant consequences in various fields, from economic inflation to ecological population booms. It's a mathematical concept with profound real-world implications, modeling scenarios where quantities multiply over time.

Summary of Function Values:

To summarize our calculations for $f(x) = 2e^x$:

• $f(-3) = 2e^{-3} \approx 0.099574$
• $f(-1) = 2e^{-1} \approx 0.73576$
• $f(0) = 2e^0 = 2$
• $f(1) = 2e^1 \approx 5.43656$
• $f(3) = 2e^3 \approx 40.171$

These values clearly demonstrate the behavior of the exponential function $f(x) = 2e^x$: it approaches $0$ as $x$ approaches negative infinity, passes through the y-intercept $(0, 2)$, and grows rapidly as $x$ approaches positive infinity. Understanding these specific calculations provides a solid foundation for comprehending more complex exponential models and their applications.

For more insights into exponential functions and their properties, you can explore resources from Khan Academy on exponential functions, or delve deeper into the mathematical constant $e$ on the Wikipedia page for Euler's number.