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

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Hey guys! Let's dive into the fascinating world of exponential functions and figure out the range of a specific one. We're going to break down the function f(x) = (3/4)^x - 4 step by step, making sure you understand the key concepts involved. So, grab your thinking caps, and let's get started!

Understanding Exponential Functions

Before we can pinpoint the range of our function, it's crucial to have a solid grasp of what exponential functions are all about. An exponential function generally takes the form of f(x) = a^x, where a is a positive constant (and not equal to 1) known as the base, and x is the variable exponent. The behavior of these functions hinges significantly on the value of the base a.

When the base a is greater than 1 (a > 1), the function represents exponential growth. This means as x increases, the value of f(x) increases dramatically, shooting towards infinity. Conversely, as x decreases (becomes more negative), f(x) gets closer and closer to zero, approaching it asymptotically. Think of functions like 2^x or 10^x – they climb rapidly as x moves to the right on the graph.

Now, when the base a lies between 0 and 1 (0 < a < 1), we're dealing with exponential decay. In this scenario, as x increases, the function's value f(x) decreases, inching closer to zero. On the flip side, as x decreases, f(x) soars towards infinity. Our function, (3/4)^x, perfectly exemplifies this decay behavior, since its base (3/4) is nestled between 0 and 1. The exponential decay is a critical aspect in determining the range.

Key Characteristics of Exponential Functions

Let's highlight some key characteristics that will help us determine the range of f(x):

  • Horizontal Asymptote: Exponential functions of the form a^x (where a is positive and not equal to 1) have a horizontal asymptote at y = 0. This means the function's graph gets incredibly close to the x-axis (y = 0) but never actually touches or crosses it. It's like an invisible boundary the function dances around.
  • Domain: The domain of an exponential function a^x is all real numbers. You can plug in any value for x, whether it's positive, negative, zero, a fraction, or an irrational number, and the function will happily churn out a result. There are no forbidden x-values here!
  • Range: For a basic exponential function a^x (where a is positive and not equal to 1), the range is all positive real numbers. In fancy mathematical notation, we write this as (0, ∞). The function's output will always be a positive number; it can be infinitesimally small, but never zero or negative. Understanding this is fundamental to figuring out the range of more complex variations.

Analyzing Our Function: f(x) = (3/4)^x - 4

Okay, with the fundamentals of exponential functions firmly in place, let's turn our attention to our specific function: f(x) = (3/4)^x - 4. We've got a base of 3/4, which, as we discussed, indicates exponential decay. But there's also that “- 4” hanging around. This is a vertical shift, and it plays a vital role in determining the range.

The Impact of the Vertical Shift

That “- 4” isn't just sitting there idly; it's causing a vertical shift in our graph. Imagine the basic exponential decay function (3/4)^x. It has a horizontal asymptote at y = 0, and its range is all positive real numbers (0, ∞). Now, when we subtract 4 from the entire function, we're effectively taking every point on the graph and dragging it down 4 units. This transformation has a direct impact on the range.

So, what happens to our horizontal asymptote? It also gets shifted down 4 units, from y = 0 to y = -4. This is critical! The function f(x) = (3/4)^x - 4 will now approach y = -4 as x heads towards infinity, but it will never actually reach it. The line y = -4 becomes our new invisible boundary.

Determining the Range

Now we're ready to pinpoint the range. We know the function f(x) = (3/4)^x itself produces only positive values. When we subtract 4, we're essentially taking those positive values and shifting them down. The smallest value (3/4)^x can approach is 0 (though it never actually gets there). So, the smallest value f(x) can approach is 0 - 4 = -4.

Since the exponential part (3/4)^x is always greater than 0, f(x) = (3/4)^x - 4 will always be greater than -4. It can get incredibly close to -4, but it will never touch it or dip below it. Therefore, the range of our function is all real numbers greater than -4.

Expressing the Range Mathematically

We've figured out the range conceptually, but let's express it using mathematical notation. There are a couple of ways we can do this:

  • Set-builder notation: This is a concise way to define a set based on a rule. In this case, we can write the range as {y | y > -4}. This reads as