Exploring F(x) = (1/10)^x Understanding Exponential Function Properties

Hey guys! Let's dive into the world of exponential functions and explore the characteristics of a specific one: f(x) = (1/10)^x. We'll break down its range, domain, key points on its graph, and whether it's an increasing or decreasing function. So, buckle up and let's get started!

Understanding Exponential Functions

Before we tackle the specifics of f(x) = (1/10)^x, let's refresh our understanding of exponential functions in general. An exponential function has the form f(x) = a^x, where a is a positive constant called the base, and x is the exponent. The behavior of an exponential function is heavily influenced by the value of the base a. If a is greater than 1, the function represents exponential growth. If a is between 0 and 1, the function represents exponential decay. This fundamental distinction plays a crucial role in determining the function's properties, including its range, domain, and whether it's increasing or decreasing. Understanding the base is key to unlocking the secrets of exponential functions, and it forms the foundation for analyzing specific examples like the one we're about to explore.

Key Characteristics of Exponential Functions

Exponential functions possess unique characteristics that set them apart from other types of functions. One crucial aspect is their asymptotic behavior. An asymptote is a line that the graph of a function approaches but never quite touches. Exponential functions often have horizontal asymptotes, which influence their range. The domain of an exponential function, the set of all possible input values (x), is typically all real numbers. However, the range, the set of all possible output values (y), is restricted depending on the base a. When the base is between 0 and 1, as in our example, the function's values get progressively smaller as x increases, but they never actually reach zero. This leads to a horizontal asymptote at y = 0 and restricts the range. Furthermore, exponential functions exhibit either exponential growth or decay, depending on the base. These fundamental characteristics are essential for understanding and analyzing exponential functions effectively. By grasping these key properties, we can better predict and interpret the behavior of these functions in various contexts.

1. Range of f(x) = (1/10)^x

Let's talk about the range of our function, f(x) = (1/10)^x. Guys, the range of a function is basically all the possible y-values (or output values) that the function can produce. When we're dealing with exponential functions, we need to consider what happens as x gets really big (positive) and really small (negative).

Think about it: As x gets larger and larger, (1/10)^x gets smaller and smaller, approaching zero. It never actually reaches zero, though. No matter how big x gets, (1/10)^x will always be a tiny positive number. Now, what happens when x becomes a very large negative number? (1/10)^x becomes a very large positive number. So, the function can take on any positive value. This means the y-values can be anything greater than zero. Therefore, the range of f(x) is y > 0, not y > 1/10. So, option A is incorrect. Understanding the behavior of exponential functions as x approaches positive and negative infinity is crucial for determining the range. Exponential functions with a base between 0 and 1, like our example, exhibit a decreasing trend and approach zero asymptotically. This means they never actually reach zero, but they get infinitesimally close. As x becomes increasingly negative, the function values grow without bound, highlighting the absence of an upper limit. Analyzing the function's behavior in these extreme cases provides a clear picture of the range, allowing us to accurately define the set of all possible output values.

2. Graph Contains (1, 1/10)

Now let's see if the graph of f(x) contains the point (1, 1/10). To check this, we simply substitute x = 1 into our function and see what we get for f(x).

So, f(1) = (1/10)^1 = 1/10. This means when x is 1, y is 1/10. Therefore, the point (1, 1/10) does lie on the graph of f(x). Option B is correct! This straightforward substitution method is a fundamental technique for verifying whether a particular point lies on the graph of a function. By plugging in the x-coordinate and calculating the corresponding y-coordinate, we can confirm if the point satisfies the function's equation. In the context of exponential functions, this process is particularly useful for identifying key points and understanding the function's behavior. For instance, knowing that (1, 1/10) lies on the graph of f(x) = (1/10)^x provides a tangible reference point for visualizing the function's decreasing trend and its asymptotic approach to the x-axis. Furthermore, this method highlights the direct relationship between input and output values, reinforcing the core concept of function evaluation.

3. Domain of f(x) = (1/10)^x

The domain of a function refers to all the possible x-values (input values) that we can plug into the function without causing any mathematical mayhem. In other words, what values of x are allowed?

For exponential functions like f(x) = (1/10)^x, there aren't really any restrictions on what x can be. We can raise (1/10) to any power – positive, negative, zero, even fractions or decimals! This means the domain of f(x) is all real numbers, not just x > 0. Therefore, option C is incorrect. The absence of restrictions on the input values for exponential functions stems from the fundamental properties of exponentiation. Unlike certain functions, such as square roots or logarithms, exponential functions do not encounter undefined results for specific x-values. This unconstrained domain allows for a wide range of inputs, which contributes to the function's versatility and applicability in various mathematical and real-world contexts. Recognizing the domain as all real numbers is essential for accurately interpreting the behavior of exponential functions and understanding their relationships to other mathematical concepts.

4. Increasing or Decreasing?

Finally, let's figure out whether f(x) = (1/10)^x is an increasing or decreasing function. An increasing function goes up as x increases, while a decreasing function goes down as x increases.

Remember what we said earlier about the base of the exponential function? In our case, the base is 1/10, which is between 0 and 1. When the base is between 0 and 1, the exponential function is always decreasing. This is because as we increase x, we're essentially taking smaller and smaller fractions of 1. So, f(x) is decreasing, not increasing. Option D is incorrect. The relationship between the base of an exponential function and its increasing or decreasing behavior is a cornerstone of understanding these functions. When the base lies between 0 and 1, the function embodies exponential decay, characterized by a continuous decline in y-values as x increases. This decreasing trend can be visually represented by a graph that slopes downward from left to right. Conversely, when the base is greater than 1, the function exhibits exponential growth, where y-values increase exponentially as x increases. This fundamental distinction between decay and growth is crucial for modeling various real-world phenomena, from radioactive decay to population growth. By recognizing the base as the key determinant of the function's increasing or decreasing nature, we can accurately predict and interpret the behavior of exponential functions in diverse applications.

Conclusion

So, after carefully analyzing the given statements, we've determined that the only true statement for f(x) = (1/10)^x is that its graph contains the point (1, 1/10). We've explored the range, domain, and increasing/decreasing nature of this exponential function. Hopefully, this deep dive has clarified things for you guys! Remember to always consider the base when analyzing exponential functions – it's the key to unlocking their secrets! Understanding the properties of exponential functions is essential for tackling more advanced mathematical concepts and applying these functions to real-world problems. By delving into the range, domain, and increasing/decreasing nature of these functions, we gain a comprehensive understanding of their behavior and characteristics. This knowledge empowers us to analyze and interpret exponential relationships in various contexts, from financial modeling to scientific research. Furthermore, mastering these fundamental concepts lays the groundwork for exploring more complex mathematical topics, such as calculus and differential equations, where exponential functions play a pivotal role. Therefore, a solid grasp of exponential function properties is not only valuable in itself but also serves as a stepping stone to advanced mathematical proficiency.