Finding The Range Of H(x) = 2^(-x) + 3 A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the fascinating world of functions, specifically exponential functions. We're going to explore the range of a particular function, h(x) = 2^(-x) + 3. If you're scratching your head wondering what the 'range' is or how to find it, don't worry! We'll break it down step-by-step in a way that's easy to understand. So, buckle up and let's get started!

Understanding the Function h(x) = 2^(-x) + 3

First, let's take a closer look at our function: h(x) = 2^(-x) + 3. This is an exponential function, which means the variable x is in the exponent. The base of the exponent is 2, and we have a negative sign in front of x, which means this is an exponential decay function. The '+ 3' at the end shifts the entire function upwards by 3 units. Grasping this fundamental structure is crucial for determining the range. To truly understand what's going on, let's consider what happens to the term 2^(-x) as x takes on different values. When x is a large positive number, -x becomes a large negative number. 2 raised to a large negative power approaches 0. For example, 2^(-10) is 1/1024, which is already quite small. As x gets even larger, 2^(-x) gets closer and closer to 0, but it never actually reaches 0. Conversely, when x is a large negative number, -x becomes a large positive number. 2 raised to a large positive power becomes a very large number. For example, 2^(10) is 1024, and it grows rapidly as the exponent increases. This exponential behavior is key to figuring out the range. We also need to consider the '+ 3' in our function. This constant term shifts the entire graph vertically upwards. Imagine the basic exponential decay function 2^(-x). It approaches the x-axis (y = 0) as x gets large. Adding 3 to the function lifts this horizontal asymptote up to y = 3. This shift is essential for determining the lower bound of our range.

Defining the Range of a Function

Okay, before we dive deeper, let's clarify what the range actually is. The range of a function is the set of all possible output values (y-values) that the function can produce. Think of it as the vertical span of the function's graph. If you were to look at the graph of the function, the range would be the set of all y-values that the graph covers. It's different from the domain, which is the set of all possible input values (x-values). To find the range, we need to consider the behavior of the function as x varies across all possible values. For exponential functions, the range is often related to the horizontal asymptote. Remember, an asymptote is a line that the graph of a function approaches but never quite touches. In our case, the horizontal asymptote plays a critical role in defining the range. The range can be expressed in different ways, such as using interval notation, set notation, or even graphically. Understanding the range helps us understand the limitations of the function. It tells us what output values are possible and what values are not. This information is invaluable in various applications of functions, from modeling real-world phenomena to solving mathematical problems. So, keeping the definition of range in mind, let's get back to our specific function and see how we can pinpoint its range.

Determining the Range of h(x) = 2^(-x) + 3

Now, let's get down to business and figure out the range of h(x) = 2^(-x) + 3. We've already established that this is an exponential decay function shifted upwards by 3 units. This is a crucial piece of information. As x approaches positive infinity, 2^(-x) approaches 0. This means that h(x) approaches 0 + 3 = 3. However, h(x) will never actually equal 3 because 2^(-x) will always be a tiny positive number, no matter how large x gets. So, 3 is a lower bound for our range, but it's not included in the range itself. On the other hand, as x approaches negative infinity, 2^(-x) becomes incredibly large. This means that h(x) also becomes incredibly large, approaching infinity. There's no upper limit to the values that h(x) can take. Combining these two observations, we can see that the range of h(x) includes all values greater than 3, but not 3 itself. We can express this in interval notation as (3, ∞). The parenthesis indicates that 3 is not included in the range, and the infinity symbol indicates that there's no upper bound. Graphically, this means that the graph of h(x) will approach the horizontal line y = 3 but never cross it. It will extend upwards indefinitely. Therefore, the range of the function h(x) = 2^(-x) + 3 is all real numbers greater than 3. This is a key result that highlights the behavior of exponential functions and the impact of vertical shifts on their range.

Expressing the Range in Different Notations

We've determined that the range of h(x) = 2^(-x) + 3 is all real numbers greater than 3. Now, let's explore how we can express this range using different notations. This is important because different situations might call for different ways of representing the same information. We've already used interval notation: (3, ∞). This is a concise way of expressing the range, where the parenthesis indicates that 3 is not included, and the infinity symbol indicates that there's no upper bound. Another way to express the range is using set notation. In set notation, we write the range as {y | y > 3}. This reads as