Finding The Range Of Y = 2e^x - 1 A Comprehensive Guide

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Finding the range of a function is a fundamental concept in mathematics, especially in calculus and precalculus. The range represents all possible output values (y-values) that the function can produce. In this article, we will delve into determining the range of the exponential function y = 2e^x - 1. We'll explore the properties of exponential functions, transformations, and how these aspects influence the range. Understanding these principles will allow us to confidently conclude the correct answer.

Exponential Functions: A Brief Overview

To understand the range of y = 2e^x - 1, we must first grasp the basics of exponential functions. An exponential function is a function in which the independent variable (x) appears in the exponent. The general form of an exponential function is y = a^x, where 'a' is a constant called the base and 'x' is the exponent. A crucial case is when the base 'a' is the mathematical constant 'e' (Euler's number), approximately equal to 2.71828. This gives us the natural exponential function, y = e^x, which is a cornerstone of mathematical analysis.

Key Properties of the Natural Exponential Function y = e^x

The natural exponential function y = e^x possesses several essential properties that dictate its behavior:

  1. Domain: The domain of y = e^x is all real numbers (-∞ < x < ∞). This means that you can input any real number as the value of x.
  2. Range: The range of y = e^x is all positive real numbers (0 < y < ∞). The function never reaches zero or becomes negative. As x approaches negative infinity, e^x approaches 0, and as x approaches positive infinity, e^x approaches infinity.
  3. Monotonicity: The function is strictly increasing, meaning that as x increases, y also increases. This is a key characteristic of exponential growth.
  4. Y-intercept: The graph of y = e^x intersects the y-axis at the point (0, 1), because e^0 = 1.
  5. Asymptote: The x-axis (y = 0) is a horizontal asymptote for the function. The graph approaches the x-axis as x tends to negative infinity but never actually touches or crosses it.

Understanding these properties of the base natural exponential function is crucial for analyzing transformations, which will allow us to determine the range of more complex exponential functions like the one given in the problem.

Transformations of Exponential Functions

Now, let’s consider the given function, y = 2e^x - 1. This function is a transformation of the basic exponential function y = e^x. Transformations involve altering the graph of a function through various operations such as stretching, compressing, reflecting, and shifting. In this case, we have two transformations applied to y = e^x:

  1. Vertical Stretch: The term '2' in front of e^x represents a vertical stretch by a factor of 2. This means that the y-values of the function are multiplied by 2. So, the function 2e^x will have y-values twice as large as e^x for any given x.
  2. Vertical Shift: The term '- 1' represents a vertical shift downward by 1 unit. This means that the entire graph is moved down 1 unit along the y-axis. Each y-value is reduced by 1.

How Transformations Affect the Range

Each transformation has a distinct impact on the range of the function. Let’s analyze the effects of each transformation step-by-step:

  1. Original function: y = e^x has a range of (0, ∞). All positive real numbers.
  2. Vertical Stretch: The transformation y = 2e^x stretches the graph vertically by a factor of 2. This affects the range by multiplying all y-values by 2. So, the new range becomes (2 * 0, 2 * ∞), which simplifies to (0, ∞). The vertical stretch does not change the fact that the function’s output remains positive but effectively makes the growth twice as rapid.
  3. Vertical Shift: The transformation y = 2e^x - 1 shifts the graph downward by 1 unit. This impacts the range by subtracting 1 from all y-values. The new range becomes (0 - 1, ∞ - 1), which simplifies to (-1, ∞). This shift is crucial because it changes the lower bound of the range, which now starts from -1 and goes to infinity.

By understanding how these transformations affect the range, we can determine the range of the transformed function y = 2e^x - 1 precisely.

Determining the Range of y = 2e^x - 1

Based on our analysis of the transformations, we can now definitively determine the range of the function y = 2e^x - 1. The function e^x has a range of (0, ∞). The vertical stretch by a factor of 2 (2e^x) maintains this range (0, ∞). However, the vertical shift down by 1 unit (- 1) changes the range to (-1, ∞).

Therefore, the range of the function y = 2e^x - 1 is all real numbers greater than -1. This means that the function can output any value greater than -1, but it will never be equal to or less than -1.

Visualizing the Range

To further solidify our understanding, we can visualize the graph of y = 2e^x - 1. Imagine starting with the basic exponential curve of y = e^x. The vertical stretch makes the curve steeper, and the vertical shift moves the entire curve down one unit. This shift is critical because the horizontal asymptote of y = e^x is the x-axis (y = 0), but the horizontal asymptote of y = 2e^x - 1 is the line y = -1. The graph approaches this line as x tends to negative infinity but never touches or crosses it. The function grows without bound as x approaches positive infinity.

This visualization provides an intuitive understanding of why the range is all real numbers greater than -1. The lowest possible y-value is infinitesimally close to -1, and the function continues to grow infinitely upwards.

Analyzing the Given Options

Now that we have determined the range of the function y = 2e^x - 1, we can evaluate the given options:

  • A. all real numbers less than -1: This is incorrect. The function's range consists of values greater than -1, not less than -1.
  • B. all real numbers greater than -1: This is the correct range we derived through our analysis of exponential function transformations.
  • C. all real numbers less than 1: This is incorrect. While the function does have y-values less than 1 for some x-values, this option does not fully capture the actual range, which starts at -1.
  • D. all real numbers greater than 1: This is incorrect. The function does have y-values greater than 1, but the range starts at -1, meaning it includes values between -1 and 1.

Therefore, the correct answer is B. all real numbers greater than -1.

Conclusion

In conclusion, we have successfully determined that the range of the function y = 2e^x - 1 is all real numbers greater than -1. This conclusion was reached by understanding the properties of exponential functions, particularly the natural exponential function y = e^x, and by analyzing the effects of transformations—vertical stretch and vertical shift—on the range. The process involved breaking down the function, identifying transformations, and considering how each transformation altered the range. Through visualization and careful analysis, we confirmed the accurate range.

Understanding the range of functions is a crucial skill in mathematics. This article provided a step-by-step guide to determining the range of an exponential function through transformations. By applying these principles, one can confidently tackle similar problems involving various types of functions and their transformations. This deeper understanding not only helps in solving problems but also in appreciating the behavior and nature of functions, which is essential in higher-level mathematical studies.

By mastering the concepts discussed here, you'll be well-equipped to handle range-related questions and build a stronger foundation in mathematics. The ability to analyze and interpret function behavior is a valuable asset in various fields that rely on mathematical modeling and analysis.