Simplifying Exponential Functions Unveiling Initial Value Base And Domain Of F(x)=7(cube Root Of 54)^x
Hey guys! Today, we're diving deep into the world of exponential functions, specifically tackling the function f(x) = 7(∛54)x. We're not just going to leave it as is; we're going to simplify it, figure out its initial value, nail down its base, and explore its domain. So, buckle up, and let's get started!
Deconstructing the Function: Simplify f(x) = 7(∛54)^x
Let's face it, f(x) = 7(∛54)x looks a bit intimidating at first glance. But don't worry, we're going to break it down step-by-step until it's something much easier to handle. The key here is to simplify the cube root of 54 (∛54). Remember, the goal is to express it in a simpler form, ideally with a smaller number under the radical. So, when we simplify exponential functions, we aim for clarity and ease of use.
The first thing we need to do is prime factorize 54. If you break it down, you'll find that 54 is equal to 2 * 3 * 3 * 3, or 2 * 33. Now, we can rewrite the cube root of 54 as ∛(2 * 33). This is where things start to get interesting. Because we have a cube (33) inside a cube root, we can pull that 3 out! So, ∛(2 * 33) becomes 3∛2. We're getting somewhere now!
Now, let's plug this back into our original function. f(x) = 7(∛54)x now transforms into f(x) = 7(3∛2)x. This is already a lot cleaner, but we can still do a bit more to make it super clear. We can use the power of a product rule, which says that (ab)x = axbx. Applying this to our function, we get f(x) = 7 * 3x * (∛2)*x.
This simplified form, f(x) = 7 * 3x * (∛2)*x, gives us a much clearer picture of what's going on. We can see the exponential growth more directly, and it will be easier to answer the questions that follow. This is why function simplification is such a crucial skill in mathematics. By making the function easier to understand, we open the door to solving a wider range of problems.
Unveiling the Initial Value of f(x) = 7(3∛2)^x
The initial value of a function is a fundamental concept, guys, and it's super easy to find! It's simply the value of the function when x is equal to 0. In other words, it's the point where the function's graph crosses the y-axis. So, to find the initial value of f(x) = 7(3∛2)x, we just need to plug in 0 for x. Finding the initial value helps us anchor the function on the graph.
Let's do it: f(0) = 7(3∛2)0. Now, remember a crucial rule of exponents: anything (except 0) raised to the power of 0 is equal to 1. So, (3∛2)0 becomes 1. Our equation simplifies to f(0) = 7 * 1, which, of course, is just 7.
Therefore, the initial value of the function f(x) = 7(3∛2)x is 7. This tells us that the function starts at the point (0, 7) on the coordinate plane. Knowing the initial value is incredibly helpful when graphing the function or analyzing its behavior. It's like having a starting point for your journey through the function's world! This initial value analysis is key to understanding the function's behavior.
Pinpointing the Simplified Base: A Closer Look at the Exponential Growth
Now, let's talk about the base of our simplified exponential function. The base is the heart of an exponential function; it determines whether the function is growing or decaying and how quickly it's changing. In our simplified form, f(x) = 7 * 3x * (∛2)*x, we need to identify the part that's being raised to the power of x. Understanding the exponential base is crucial for understanding growth patterns.
Looking at the function, we can see that we have two terms with x in the exponent: 3x and (∛2)x. To find the simplified base, we need to combine these. Remember the rule of exponents that says ax * bx = (ab)x? We can use this in reverse!
So, 3x * (∛2)x becomes (3 * ∛2)x. This means the simplified base of our exponential function is 3∛2. This single value encapsulates the core growth factor of the function. The base identification process helps us quantify the rate of change.
But wait, the question asks for the simplified base in the form of a number multiplied by ∛2. We've already got that! Our base is indeed 3∛2. This tells us that for every increase of 1 in x, the function's value is multiplied by 3∛2. That's a pretty significant growth factor, indicating a fairly rapid increase in the function's value as x increases. This simplified base determination is vital for predicting long-term function behavior.
Delving into the Domain: Where Does This Function Live?
Finally, let's talk about the domain of our function. The domain, in simple terms, is the set of all possible input values (x-values) that we can plug into the function and get a valid output (y-value). For exponential functions, the domain is usually pretty straightforward, guys, but it's always good to check. Exploring the function domain is essential for complete function understanding.
Think about it: can we plug in any number we want for x in the function f(x) = 7(3∛2)x? Can we use positive numbers? Negative numbers? Zero? Fractions? The answer is yes to all of those! There are no restrictions on the values we can use for x. We can raise any real number to a power, and we won't run into any mathematical issues like dividing by zero or taking the square root of a negative number.
Therefore, the domain of the function f(x) = 7(3∛2)x is all real numbers. We can write this in a few different ways: we can use interval notation (-∞, ∞), or we can use set-builder notation {x | x ∈ ℝ}, which means "the set of all x such that x is an element of the set of real numbers." Understanding the domain constraints ensures we're working within the function's valid input range. This domain analysis completes our exploration of the function's fundamental characteristics.
In conclusion, we've successfully simplified the function f(x) = 7(∛54)x, found its initial value, identified its simplified base, and determined its domain. By breaking down the problem step-by-step, we've transformed a seemingly complex function into something we can understand and analyze with ease. Keep practicing, guys, and you'll become exponential function masters in no time!