Finding The Range Of F(x) = (3/4)^x - 4 A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of exponential functions and figuring out their range. Specifically, we're going to tackle the function f(x) = (3/4)^x - 4. Finding the range means identifying all the possible output values (y-values) that this function can produce. Think of it like this: if you put every possible x-value into the function, what set of y-values will you get back? This is a crucial concept in understanding functions, and it pops up everywhere in mathematics, from calculus to real-world applications like modeling population growth or radioactive decay. So, let's break this down step by step and make sure we nail it!

When we talk about the range of a function, we're essentially asking, "What are all the possible y-values that this function can spit out?" For exponential functions, this question gets really interesting because of how they behave. Remember, exponential functions have a variable in the exponent, and this leads to some unique properties. One of the most important things to remember is that an exponential function of the form a^x, where a is a positive number not equal to 1, will always produce a positive result. It will never be zero or negative. This is because no matter what value you plug in for x, raising a positive number to that power will always result in a positive number. This is the foundation for understanding the range of our specific function, f(x) = (3/4)^x - 4. We need to consider how this basic exponential behavior is affected by the subtraction of 4. We'll dive into that in the next sections, exploring the key aspects of exponential functions and their ranges, and then apply that knowledge to our specific problem. So buckle up, and let's get started!

When looking at the range, we need to understand how transformations affect the basic exponential function. The "- 4" in our function f(x) = (3/4)^x - 4 is a vertical shift. It takes the basic exponential function (3/4)^x and shifts it downwards by 4 units. This is a game-changer because it directly impacts the lower bound of our range. Without the "- 4", the range would be all positive numbers. But now, we need to account for this shift. Think of it this way: the exponential part, (3/4)^x, will still produce only positive values, but then we subtract 4 from each of those values. This means our entire range will be shifted down by 4 units. This vertical shift is a key concept in understanding transformations of functions, and it's not limited to exponential functions. It applies to all sorts of functions, from quadratics to trigonometric functions. Recognizing these transformations is a powerful tool in your mathematical arsenal. So, by understanding how the "- 4" shifts the entire function downwards, we're one step closer to pinpointing the range of f(x). We're not just blindly following steps; we're building a conceptual understanding of what's happening. And that's what makes mathematics truly fascinating!

Understanding the Exponential Component (3/4)^x

Let's start by focusing on the exponential part of our function, which is (3/4)^x. This is a crucial step because the behavior of this component dictates the overall behavior of the function. Now, (3/4)^x is a bit special because the base (3/4) is a fraction between 0 and 1. This means that as x gets larger and larger, the value of (3/4)^x gets smaller and smaller, approaching zero. This is the hallmark of an exponential decay function. Conversely, as x becomes a very large negative number, (3/4)^x becomes a very large positive number. It shoots off towards infinity. This is because a negative exponent means we're dealing with the reciprocal of the base raised to the positive exponent. So, (3/4)^-x is the same as (4/3)^x, which grows rapidly as x increases. Understanding this behavior is key to grasping the range.

To really nail this down, let's think about some specific values. When x is 0, (3/4)^0 is equal to 1 (remember, anything raised to the power of 0 is 1). When x is 1, (3/4)^1 is simply 3/4. When x is 2, (3/4)^2 is 9/16, which is even smaller. See how the value is decreasing? Now, let's go in the other direction. When x is -1, (3/4)^-1 is 4/3. When x is -2, (3/4)^-2 is 16/9, which is even larger. This illustrates the exponential decay as x increases and the exponential growth as x decreases. This understanding of the exponential component is the foundation upon which we'll build our understanding of the entire function's range. We're not just memorizing rules; we're building intuition about how these functions behave. And that's what makes math fun and empowering!

The exponential function (3/4)^x will never actually reach zero. It gets infinitely close to zero as x approaches infinity, but it never quite touches it. This is a concept called a horizontal asymptote. The line y = 0 is a horizontal asymptote for the function (3/4)^x. This is a critical point because it defines the lower bound of the values that (3/4)^x can take. Since it never reaches zero, the smallest value it can get infinitely close to is zero, but it will always be a tiny bit more than zero. On the other hand, as x approaches negative infinity, (3/4)^x grows without bound, shooting off towards positive infinity. There is no upper limit to the values it can take. This means that the exponential component (3/4)^x can take on any positive value. It's this understanding that allows us to predict the ultimate range of the full function. We know the exponential part will always be positive, and now we need to consider how the “- 4” shifts everything. So, we're building a complete picture, piece by piece, until we have a solid grasp of the range. This is the power of breaking down complex problems into smaller, more manageable parts!

Accounting for the Vertical Shift

Now that we understand the behavior of the exponential component (3/4)^x, let's bring in the "- 4". Remember, this "- 4" is a vertical shift. It takes the entire graph of (3/4)^x and moves it down 4 units. This seemingly simple transformation has a profound impact on the range of the function. Since (3/4)^x always produces positive values (greater than 0), the smallest value it can get infinitely close to is 0. When we subtract 4 from this, the smallest value that f(x) = (3/4)^x - 4 can get infinitely close to is -4. This is a crucial point. The function will never actually reach -4, but it will get arbitrarily close to it. This means that -4 is the lower bound of our range.

To visualize this, imagine the graph of (3/4)^x. It's a curve that starts high on the left and gradually decreases, getting closer and closer to the x-axis (y = 0) but never touching it. Now, imagine grabbing that entire graph and sliding it down 4 units. The x-axis (y = 0) also slides down 4 units to become the line y = -4. Our curve now gets closer and closer to the line y = -4 but never touches it. This visually illustrates why -4 is the lower bound of the range. The function values will always be greater than -4. They can get as close as you like to -4, but they will never equal -4. This concept of a horizontal asymptote shifting due to a vertical translation is incredibly important. It's not just a trick to memorize; it's a fundamental understanding of how transformations affect function behavior. And it's this understanding that empowers us to tackle a wide range of mathematical problems with confidence and clarity!

As (3/4)^x can take on any positive value, when we subtract 4, the values of f(x) can be any number greater than -4. There's no upper bound because as x approaches negative infinity, (3/4)^x approaches infinity, and subtracting 4 doesn't change that unbounded nature. Therefore, the range of f(x) = (3/4)^x - 4 consists of all real numbers greater than -4. We've carefully considered both the exponential component and the vertical shift, and now we have a clear picture of the possible output values of our function. This is the essence of finding the range: understanding the individual components and how they interact to shape the overall behavior of the function. It's not about memorizing formulas; it's about building a conceptual understanding of the underlying mathematical principles. And that's what truly makes math meaningful and rewarding!

Determining the Range and Choosing the Correct Option

So, we've established that the function f(x) = (3/4)^x - 4 will always produce values greater than -4. It can get infinitely close to -4, but it will never actually reach -4. There is no upper limit to the values it can take; they can go all the way to positive infinity. This means the range of our function is all y-values such that y > -4. Now, let's look back at the answer choices and see which one matches our conclusion.

• A. {y | y > -4} This looks promising! It says the range is all y-values greater than -4, which is exactly what we found.
• B. {y | y > 3/4} This is incorrect. We know the range is based on the vertical shift of -4, not the base of the exponential function.
• C. {y | y < -4} This is the opposite of what we found. The function values are greater than -4, not less than.
• D. {y | y < 3/4} This is also incorrect for the same reason as option B. It focuses on the base of the exponential function instead of the vertical shift.

Therefore, the correct answer is A. {y | y > -4}. We've carefully analyzed the function, understood the behavior of the exponential component, accounted for the vertical shift, and now we've confidently identified the range. This is a testament to the power of a systematic approach and a solid understanding of the underlying concepts. We didn't just guess; we reasoned our way to the correct answer. And that's the key to success in mathematics!

In conclusion, by understanding the behavior of the exponential function (3/4)^x and the impact of the vertical shift of -4, we were able to confidently determine that the range of f(x) = (3/4)^x - 4 is {y | y > -4}. This exercise highlights the importance of breaking down complex problems into smaller, more manageable parts and building a solid conceptual understanding of the underlying principles. So keep practicing, keep exploring, and keep having fun with math!