Finding The Range Of Y=4e^x An Exponential Function Exploration
Hey guys! Today, we're diving into the fascinating world of exponential functions, specifically focusing on the function y = 4e^x. Our mission? To figure out its range. Now, if you're scratching your head wondering what a range is, don't worry! We'll break it down step by step. The range of a function is simply the set of all possible output values (y-values) that the function can produce. Think of it like this: you feed the function different inputs (x-values), and the range is the collection of all the results you get out. So, let's get started and uncover the range of y = 4e^x!
Understanding the Exponential Function
Before we jump into the specifics of our function, let's take a moment to understand the basic exponential function, y = e^x. This is the foundation upon which our function is built. The number 'e' is a special mathematical constant, approximately equal to 2.71828. It's an irrational number, meaning its decimal representation goes on forever without repeating, just like pi. The exponential function y = e^x has some key characteristics that are crucial for understanding its range.
First, let's think about what happens as we plug in different values for x. If x is a large positive number, e^x becomes a very large positive number as well. For example, e^10 is already a pretty big number (over 22,000!). As x gets even larger, e^x grows incredibly quickly. This is the essence of exponential growth. Now, what happens when x is zero? Well, any number raised to the power of zero is 1, so e^0 = 1. This gives us a crucial point on the graph of y = e^x: the point (0, 1). This is where the graph intersects the y-axis.
But what about negative values of x? This is where things get interesting. When x is negative, e^x becomes a fraction. Remember that e^-x is the same as 1/e^x. So, as x becomes a large negative number, e^x gets closer and closer to zero, but it never actually reaches zero. This is a crucial point! The exponential function y = e^x approaches the x-axis (y = 0) as x goes towards negative infinity, but it never touches or crosses it. This behavior is described as having a horizontal asymptote at y = 0.
The graph of y = e^x visually represents these characteristics. It starts very close to the x-axis on the left side (for large negative x), then it rises slowly at first, passing through the point (0, 1), and then it shoots up dramatically as x increases. The graph is always above the x-axis, reflecting the fact that e^x is always positive. This is a key takeaway: e^x is always greater than 0 for any real number x. This understanding is fundamental to determining the range of our target function, y = 4e^x.
Analyzing y = 4e^x
Okay, now that we've got a solid grasp of the basic exponential function y = e^x, let's shift our focus to our specific function: y = 4e^x. This function is a slight modification of the basic exponential function, and understanding how this modification affects the graph is key to figuring out the range. Notice that the only difference between y = e^x and y = 4e^x is the multiplication by 4. This seemingly small change has a significant impact on the function's behavior.
Multiplying a function by a constant like 4 is called a vertical stretch. It essentially stretches the graph of the function vertically away from the x-axis. In this case, every y-value of the original function y = e^x is multiplied by 4 to get the corresponding y-value of y = 4e^x. This means that the graph of y = 4e^x will be steeper than the graph of y = e^x. It will still have the same basic shape, but it will be stretched upwards.
Let's think about how this vertical stretch affects some key points. We know that e^0 = 1, so for the function y = 4e^x, when x = 0, we have y = 4 * e^0 = 4 * 1 = 4. So, the graph of y = 4e^x passes through the point (0, 4), instead of (0, 1) like the basic exponential function. This is a direct consequence of the vertical stretch. Now, let's consider the behavior as x approaches negative infinity. We know that e^x approaches 0 as x goes towards negative infinity. Therefore, 4e^x will also approach 0 as x goes towards negative infinity. However, just like e^x, 4e^x will never actually reach 0.
This is a crucial observation! Because e^x is always greater than 0, multiplying it by a positive number like 4 will still result in a positive number. Therefore, 4e^x is always greater than 0 for any real number x*. This means the graph of y = 4e^x will always be above the x-axis. It will get closer and closer to the x-axis as x becomes a large negative number, but it will never touch or cross it. This reinforces the idea that y = 0 is a horizontal asymptote for this function as well.
Determining the Range
Alright, we've dissected the function y = 4e^x and understand its behavior. Now, we're ready to pinpoint its range. Remember, the range is the set of all possible output (y) values. We've established that 4e^x is always greater than 0. This means that the function can produce any positive number as an output. As x becomes a large positive number, 4e^x becomes an incredibly large positive number as well. There's no upper limit to the y-values the function can produce.
However, we also know that 4e^x never actually reaches 0. It gets infinitesimally close, but it never touches it. So, 0 is not included in the range. This is a subtle but important distinction. Therefore, the range of the function y = 4e^x is all real numbers greater than 0. We can express this mathematically as y > 0 or using interval notation as (0, ∞). This means that the function can take on any positive value, but it cannot be equal to zero or any negative value.
Let's revisit the answer choices to confirm our understanding. We've determined that the range is all real numbers greater than 0. Looking at the options, we can confidently say that the correct answer is A. all real numbers greater than 0. The other options are incorrect because they either include negative numbers, zero, or an upper bound, none of which apply to the range of y = 4e^x.
So, there you have it! We've successfully navigated the world of exponential functions and determined the range of y = 4e^x. Remember, understanding the basic exponential function and how transformations affect it is crucial for tackling these types of problems. Keep practicing, and you'll become a range-finding pro in no time! Let's recap the key concepts. We emphasize that the range of an exponential function in the form y = ae^x where a > 0 is always all real numbers greater than 0. This knowledge can be directly applied to solve similar problems efficiently.
Conclusion
In conclusion, determining the range of y = 4e^x involves understanding the fundamental properties of exponential functions and how transformations affect their behavior. The basic exponential function, y = e^x, is always positive, and multiplying it by a positive constant like 4 simply stretches the graph vertically without changing the fact that it remains strictly above the x-axis. Therefore, the range of y = 4e^x is all real numbers greater than 0. This exploration not only helps in solving specific problems but also enhances the overall understanding of exponential functions, which are vital in various fields of mathematics and science. Remember, the key is to break down the problem, understand the core concepts, and apply them logically. Great job, guys, we nailed it!
Practice Questions
To further solidify your understanding, try determining the range of the following functions:
- y = 2e^x
- y = 0.5e^x
- y = -e^x (Hint: Consider what happens when you multiply by a negative number)
Good luck, and happy problem-solving!