Range Of Exponential Function Y=4e^x Explained

In mathematics, determining the range of a function is a fundamental concept. The range represents the set of all possible output values (y-values) that a function can produce. For the function y = 4e^x, we aim to identify the range, which will reveal the spectrum of y-values this function can generate. This exploration will not only provide the answer but also deepen the understanding of exponential functions and their behavior. Before diving into the specifics of y = 4e^x, it is crucial to grasp the nature of the basic exponential function, e^x, where 'e' is the base of the natural logarithm, approximately equal to 2.71828. The exponential function e^x is defined for all real numbers (x), meaning x can take any value from negative infinity to positive infinity. However, the output of e^x is always positive. As x approaches negative infinity, e^x approaches 0 but never actually reaches it. As x approaches positive infinity, e^x grows without bound, heading towards infinity. Therefore, the range of the basic exponential function e^x is all positive real numbers, or (0, ∞) in interval notation. This foundational understanding is essential as we proceed to analyze the function y = 4e^x.

Analyzing the Function y=4e^x

The function y = 4e^x is a transformation of the basic exponential function e^x. The multiplication by 4 is a vertical stretch, meaning it scales the output of e^x by a factor of 4. To determine the range of y = 4e^x, we need to consider how this vertical stretch affects the possible output values. As we've established, e^x always produces positive values. When we multiply these positive values by 4, the result remains positive. The function 4 e^x will also approach 0 as x approaches negative infinity, but it will never reach 0. Similarly, as x approaches positive infinity, 4 e^x will grow without bound, heading towards infinity. Therefore, the range of y = 4e^x is all positive real numbers greater than 0. This can be expressed in interval notation as (0, ∞). In simpler terms, the function can produce any positive value, but it will never produce 0 or any negative value. The vertical stretch by a factor of 4 simply expands the range of possible outputs, but it doesn't change the fundamental characteristic of the exponential function, which is to produce only positive values. This understanding of vertical stretching and its effect on the range of a function is crucial in analyzing various transformations of functions in mathematics.

Detailed Explanation of the Range

To further solidify our understanding, let's delve into a more detailed explanation of why the range of y = 4e^x is all positive real numbers greater than 0. The exponential function e^x is inherently positive because 'e' raised to any power, whether positive, negative, or zero, will always yield a positive result. e is approximately 2.71828, a positive number, and raising a positive number to any real power will result in a positive number. The key is that e^x approaches 0 as x goes towards negative infinity, but it never actually reaches 0. This is a critical concept in understanding the behavior of exponential functions. Multiplying e^x by 4 in the function y = 4e^x simply scales the output. It doesn't change the fundamental characteristic of the function being positive. The values are stretched vertically by a factor of 4, but the lower bound remains infinitesimally close to 0. This means the function can produce values arbitrarily close to 0, but it will never actually output 0. On the other end of the spectrum, as x goes towards positive infinity, e^x grows without bound. Multiplying this unbounded growth by 4 also results in unbounded growth. The function y = 4e^x can produce infinitely large positive values. Therefore, the range encompasses all positive real numbers greater than 0, excluding 0 itself. This detailed analysis emphasizes the importance of understanding the underlying principles of exponential functions and how transformations affect their behavior and range.

Correct Answer

Based on our analysis, the correct answer is:

A. all real numbers greater than 0

This confirms that the function y = 4e^x can produce any positive real number but cannot produce 0 or any negative number. The understanding of exponential functions and their transformations is crucial for accurately determining their ranges.

Why Other Options are Incorrect

To ensure a comprehensive understanding, let's discuss why the other options provided are incorrect:

• B. all real numbers less than 0: This is incorrect because, as we've established, the exponential function e^x always produces positive values. Multiplying by 4 does not change this fundamental characteristic. The function y = 4e^x will never produce a negative output.
• C. all real numbers less than 4: This option is also incorrect. While the coefficient 4 might suggest a bound of 4, the exponential function e^x can grow without bound as x approaches positive infinity. Therefore, y = 4e^x can produce values greater than 4.
• D. all real numbers greater than 4: This option is partially correct in that the function does produce values greater than 4. However, it is incomplete. The function can produce any positive real number, including those between 0 and 4. The range is not limited to values greater than 4 only.

Understanding why these options are incorrect reinforces the concept of the range of an exponential function and how it is affected by transformations. The key takeaway is that exponential functions of the form ae^x*, where a is a positive constant, will always have a range of all positive real numbers greater than 0.

Conclusion: Mastering Exponential Function Ranges

In conclusion, determining the range of a function like y = 4e^x requires a solid understanding of exponential functions and how they behave. The range of the function y = 4e^x is all real numbers greater than 0, represented as (0, ∞) in interval notation. This is because the basic exponential function e^x always produces positive values, and multiplying by a positive constant like 4 simply scales the output without changing its fundamental positivity. We've explored why the other options are incorrect, reinforcing the concept of the range of an exponential function. Mastering these concepts is crucial for success in mathematics, particularly in calculus and analysis, where exponential functions play a significant role. By understanding the core principles and how transformations affect functions, one can confidently determine the range of a wide variety of functions. The exponential function, with its unique properties, is a cornerstone of mathematical analysis, and a thorough grasp of its behavior is invaluable. This exploration has not only provided the correct answer but also aimed to deepen the understanding of exponential functions and their ranges, equipping readers with the knowledge and confidence to tackle similar problems in the future.