Range Of Transformed Exponential Functions Explained

Hey guys! Let's dive into the fascinating world of functions and transformations. Today, we're going to tackle a problem that combines the power of exponential functions with the elegance of transformations. We'll be working with the function $f(x) = 10^x$ and its transformed cousin, $g(x) = -2f(x) + 1$. Our mission, should we choose to accept it, is to determine the range of this transformed function $g(x)$. Buckle up, because we're about to embark on a mathematical adventure!

Understanding the Exponential Foundation: f(x) = 10^x

Before we can conquer the transformed function, we need a solid understanding of the original exponential function, $f(x) = 10^x$. This function is a classic example of exponential growth. Let's break down its key characteristics:

• The Base: The base of our exponential function is 10, which is greater than 1. This signifies that as x increases, the function's value grows exponentially. Think of it like compound interest – the larger your principal, the faster your money grows!
• The Domain: The domain of $f(x) = 10^x$ is all real numbers. This means we can plug in any value for x, whether it's positive, negative, zero, or a fraction, and the function will happily churn out a result. There are no input restrictions here!
• The Range: This is where things get interesting. The range of $f(x) = 10^x$ is all positive real numbers, or $(0, \infty)$. Why is this the case? Well, no matter what value we plug in for x, the result of $10^x$ will always be greater than zero. Even if x is a large negative number, like -1000, $10^{-1000}$ is a tiny positive number (1 divided by 10 raised to the power of 1000), but it's still positive! The function approaches zero as x approaches negative infinity, but it never actually reaches zero.
• The Graph: The graph of $f(x) = 10^x$ is a smooth curve that starts very close to the x-axis on the left side (for negative x values) and then shoots upwards rapidly as x increases. It's a visual representation of exponential growth – slow at first, then explosive!

Think of the exponential function $f(x) = 10^x$ as the foundation upon which we'll build our understanding of the transformed function $g(x)$. We need to grasp its fundamental properties, especially its range, to decipher the impact of the transformations.

Now, let's consider some specific examples to solidify our understanding. If we plug in x = 0, we get $f(0) = 10^0 = 1$. If we plug in x = 1, we get $f(1) = 10^1 = 10$. If we plug in x = -1, we get $f(-1) = 10^{-1} = 0.1$. Notice how the function values are always positive, confirming our understanding of the range. We also observe the rapid growth as x increases, a hallmark of exponential functions. It is crucial to have a clear mental image of this exponential behavior to effectively analyze transformations.

Understanding the range of the parent function, $f(x) = 10^x$, is the key to unlocking the range of its transformations. The range (0, ∞) tells us that the output of the exponential part is always positive. This positivity is what we'll manipulate when we apply transformations like reflections and vertical shifts. It's like knowing the raw material before you start crafting it into a finished product. Without this baseline understanding, it's easy to get lost in the transformations themselves. So, remember, the exponential foundation is always positive!

Decoding the Transformation: g(x) = -2f(x) + 1

Now that we have a firm grasp on the original function, $f(x)$, let's turn our attention to the transformed function, $g(x) = -2f(x) + 1$. This function is a result of applying a series of transformations to $f(x)$. To find the range of $g(x)$, we need to carefully analyze the effects of each transformation.

Let's break down the transformations step by step:

1. Vertical Stretch/Compression and Reflection: The term "-2" in front of $f(x)$ represents two transformations in one. The "2" signifies a vertical stretch by a factor of 2, meaning the function's values are doubled. The negative sign indicates a reflection across the x-axis, flipping the function vertically. Imagine the graph of $f(x)$ being stretched upwards and then flipped upside down – that's the combined effect of this transformation. Initially, the range of $f(x)$ is (0, ∞). After multiplying by 2, the range becomes (0, ∞) * 2 = (0, ∞). Then, reflecting across the x-axis changes the sign, making the range (-∞, 0).
2. Vertical Shift: The "+ 1" at the end of the expression represents a vertical shift upwards by 1 unit. This means the entire graph of the function is lifted one unit in the positive y-direction. If we consider the range after the vertical stretch and reflection, which is (-∞, 0), adding 1 shifts the entire range upwards by 1 unit. Therefore, the new range becomes (-∞ + 1, 0 + 1), which simplifies to (-∞, 1).

By understanding the individual effects of each transformation, we can piece together the overall transformation and determine the final range. The vertical stretch and reflection flipped the positive range of the exponential function to negative infinity, and the vertical shift lifted the upper bound by one unit. So, the combination of these transformations gives us the final range of (-∞, 1). It is essential to understand that transformations are applied sequentially, and each transformation builds upon the previous one.

To further illustrate this, let's consider a specific point on the original function. For example, when x = 0, f(0) = 1. Now, let's see how this point transforms in g(x): g(0) = -2 * f(0) + 1 = -2 * 1 + 1 = -1. This simple calculation shows us how a point with a positive y-value in f(x) becomes a negative y-value in g(x) due to the reflection and stretch, and then gets shifted upwards by 1 unit. This point-by-point analysis can be very helpful in visualizing the overall transformation. Understanding the order of operations in transformations is paramount. We first apply stretches and reflections and then shifts. This order ensures that we accurately track the impact of each transformation on the function's range.

Determining the Range of g(x): The Final Answer

After carefully analyzing the transformations applied to $f(x)$, we've arrived at the range of $g(x) = -2f(x) + 1$. The vertical stretch and reflection caused the range to become negative, extending to negative infinity. The vertical shift then lifted the range by 1 unit. Therefore, the range of $g(x)$ is $(-\infty, 1)$.

This means that the function $g(x)$ can take on any value less than 1, but it will never reach or exceed 1. The upper bound of the range is 1, but it's not included, which is why we use a parenthesis instead of a bracket in the interval notation. The negative infinity indicates that there's no lower bound – the function can decrease without limit.

Therefore, the correct answer from the options provided is B. $(-\infty, 1)$. We've successfully navigated the transformations and pinpointed the range of the transformed function. It's like solving a puzzle where each transformation is a piece, and the final range is the completed picture.

To reiterate, the process involved understanding the base exponential function, identifying the transformations applied, and then sequentially determining the effect of each transformation on the range. This systematic approach is critical for handling any function transformation problem. It's not just about memorizing rules; it's about understanding the underlying principles of how functions behave and how transformations alter that behavior. Think of it as learning the mechanics of a machine – once you understand how the gears and levers work, you can predict the machine's behavior under different conditions.

Let's consider a graphical representation to solidify our understanding. If we were to plot the graph of $g(x)$, we would see a curve that starts very close to y = 1 on the left side (for negative x values) and then plunges downwards towards negative infinity as x increases. This visual confirmation reinforces our calculated range and provides a powerful way to check our work. It's like having a visual aid to complement your analytical skills. The graph is a tangible representation of the mathematical concept, making it easier to grasp and remember.

Key Takeaways and Further Exploration

Congratulations, guys! You've successfully conquered the challenge of finding the range of a transformed exponential function. Let's recap the key takeaways from our mathematical journey:

• Understanding the Parent Function: Always start by understanding the base function, in this case, $f(x) = 10^x$. Its properties, especially its range, are crucial for analyzing transformations.
• Decoding Transformations: Break down the transformations step by step, identifying stretches, compressions, reflections, and shifts. Analyze the impact of each transformation on the range.
• Sequential Application: Apply transformations in the correct order – stretches and reflections before shifts. This ensures accurate tracking of the range changes.
• Range Notation: Express the range using proper interval notation, distinguishing between inclusive (brackets) and exclusive (parentheses) endpoints.
• Graphical Verification: Whenever possible, visualize the function's graph to confirm your calculated range.

This problem is a stepping stone to a deeper understanding of function transformations. You can apply these same principles to other types of functions, such as quadratic, trigonometric, and logarithmic functions. The core concepts remain the same – identify the transformations and analyze their impact on the function's properties.

For further exploration, consider these questions:

• How would the range change if the vertical shift was downwards instead of upwards?
• What if we had a horizontal stretch or compression? How would that affect the range?
• Can you create your own transformed exponential function and determine its range?

The world of functions and transformations is vast and fascinating. Keep exploring, keep questioning, and keep pushing your mathematical boundaries. Remember, the journey of learning is just as rewarding as the destination!

So, there you have it! We've successfully unraveled the mysteries of the transformed exponential function and its range. Keep practicing, and you'll become a transformation master in no time!