Range Of Y=2e^x-1: A Step-by-Step Explanation

Hey guys! Let's dive into the fascinating world of functions, specifically exponential functions. Today, we're tackling a classic problem: finding the range of the function y = 2e**x - 1. This might sound intimidating at first, but trust me, we'll break it down step by step so it becomes crystal clear. We're not just going to find the answer; we're going to understand why it's the answer.

Understanding the Exponential Function

To really nail the range of y = 2e**x - 1, we first need to get cozy with the basic exponential function, y = e**x. Think of e as a special number, approximately 2.71828. It's like the π of exponential functions – a fundamental constant. Now, e**x means e raised to the power of x. What happens as x changes? Let's explore.

• When x is a large positive number: e**x becomes incredibly large. Imagine 2.71828 multiplied by itself many times – the result shoots up rapidly.
• When x is zero: e0 is equal to 1. This is a crucial point.
• When x is a large negative number: e**x gets closer and closer to zero, but it never actually reaches zero. Think of it as 1 divided by e raised to a large positive power – the denominator becomes huge, making the whole fraction tiny.

This behavior is key to understanding the range. The exponential function e**x can take on any positive value, but it never becomes zero or negative. It's always dancing just above the x-axis. Graphically, it's a curve that starts very close to the x-axis on the left and shoots up dramatically as you move to the right.

Transformations: The Key to Unlocking the Range

Now that we've got e**x down, let's look at our function: y = 2e**x - 1. See those extra bits? They're called transformations, and they shift and stretch the basic exponential function, changing its range. We have two transformations here:

1. Multiplication by 2 (2*ex*): This is a vertical stretch. It takes every y-value of the original *ex* function and multiplies it by 2. So, instead of the function approaching 0, 2e**x still approaches 0, but it does so while being vertically stretched. All positive values are now doubled, but it still only occupies positive real numbers.
2. Subtraction of 1 (2*ex* - 1): This is a vertical shift. It moves the entire graph down by 1 unit. This is the crucial transformation that affects the range. Remember, 2*ex* was always greater than 0. Subtracting 1 means our new function, 2e**x - 1, will always be greater than -1. This is because the entire graph, which previously hovered above zero, has now been dragged down one unit.

Visualizing the Transformation

Imagine the graph of e**x. It starts close to the x-axis and shoots upwards. Now, picture stretching it vertically by a factor of 2. The shape is the same, but it's taller. Finally, imagine grabbing the whole thing and sliding it down one unit. The entire curve is now shifted downwards. The horizontal asymptote, which was the x-axis (y=0), is now the line y=-1. This visualization is super helpful in understanding why the range is what it is.

Putting It All Together: Finding the Range

So, let's recap. We started with e**x, which has a range of all positive real numbers (greater than 0). Multiplying by 2 didn't change the range in terms of negative values; it still includes all positive real numbers. But subtracting 1 shifted the entire graph down, changing the lower bound. The function 2e**x - 1 will always be greater than -1. It can get infinitely close to -1, but it will never actually reach it. On the other end, as x gets larger, 2e**x - 1 grows without bound, taking on any value greater than -1.

Therefore, the range of the function y = 2e**x - 1 is all real numbers greater than -1. This corresponds to answer choice B. We've not only found the answer, but we've also understood the transformations and the behavior of the exponential function that lead us to it. Remember, understanding the why is just as important as finding the answer!

Why the Other Options Are Incorrect

It's essential not just to know the right answer, but also to understand why the other options are wrong. This helps solidify your understanding of the concept and avoid similar mistakes in the future. Let's break down why options A, C, and D are incorrect in this case.

• A. All real numbers less than -1: This is incorrect because the exponential function e**x is always positive. Multiplying by 2 doesn't change that, and subtracting 1 only shifts the lower bound to -1. The function never goes below -1.
• C. All real numbers less than 1: While the function does take on values less than 1 (for example, when x is 0, y is 1), it doesn't take on all values less than 1. It's bounded below by -1.
• D. All real numbers greater than 1: This is incorrect because the function's values can be less than 1, especially when x is negative. For instance, if x = 0, y = 2e0 - 1 = 2 - 1 = 1. However, the crucial part is that it can take values between -1 and 1, which this option misses.

By analyzing why these options are wrong, we reinforce our understanding of the range and the transformations involved. It’s a valuable practice in problem-solving!

Mastering Range Problems: Tips and Tricks

Finding the range of a function can be tricky, but with the right approach, you can ace these problems. Here are some tips and tricks to help you master range problems:

1. Understand Basic Functions: Get familiar with the graphs and ranges of basic functions like linear, quadratic, exponential, logarithmic, and trigonometric functions. Knowing these