Finding G(11) Exponential Function G(x) = Ab^x

Hey math enthusiasts! Today, we're diving into the fascinating world of exponential functions. We've got a function defined as g(x) = ab^x, and we're given two points on this function: g(6) = 11 and g(9) = 704. Our mission, should we choose to accept it (and we definitely do!), is to find the value of g(11). So, grab your thinking caps, and let's unravel this mathematical mystery!

Unveiling the Exponential Function g(x) = ab^x

Let's start by dissecting what an exponential function actually is. In its simplest form, g(x) = ab^x represents a relationship where the output (g(x)) changes exponentially with respect to the input (x). The key players here are a and b. The coefficient a acts as a vertical stretch or compression factor, essentially determining the initial value or starting point of the function. Think of it as the y-intercept if the function were plotted on a graph. The base b is where the exponential magic happens. It dictates the rate of growth (if b > 1) or decay (if 0 < b < 1) of the function. The exponent x is the variable that controls how many times we multiply b by itself, leading to the exponential behavior. Understanding the roles of a and b is crucial for predicting the function's behavior and, in our case, for finding the elusive value of g(11). The exponential function is a fundamental concept in mathematics, with applications ranging from population growth and compound interest to radioactive decay and the spread of diseases. Its unique properties make it a powerful tool for modeling real-world phenomena that exhibit rapid growth or decline. When working with exponential functions, it's essential to remember that the base b must be a positive number not equal to 1. If b were equal to 1, the function would simply be a constant function (g(x) = a), and if b were negative, the function would exhibit oscillating behavior, which is not characteristic of a typical exponential function. In our specific problem, we're given two points on the curve, which allows us to create a system of equations and solve for the unknowns a and b. Once we have determined the values of a and b, we can then confidently calculate g(11) by simply plugging in x = 11 into the function. This process highlights the importance of understanding the relationship between the parameters of an exponential function and its behavior.

Cracking the Code: Using g(6) = 11 and g(9) = 704

Now, let's put our detective hats on and use the clues we've been given. We know that g(6) = 11 and g(9) = 704. This means when we plug in x = 6 into our function, we get 11, and when we plug in x = 9, we get 704. This translates into two juicy equations:

1. 11 = ab⁶
2. 704 = ab⁹

We've got a system of two equations with two unknowns (a and b). Time to put our algebra skills to the test! A common strategy for solving such systems, especially when dealing with exponential terms, is to use substitution or elimination. In this case, elimination, specifically division, seems like a promising approach. Why? Because if we divide the second equation by the first equation, the a terms will cancel out, leaving us with an equation solely in terms of b. This simplifies the problem significantly and allows us to isolate and solve for the base of the exponential function. The process of dividing the equations is a clever trick that leverages the properties of exponents. When we divide ab⁹ by ab⁶, we are essentially subtracting the exponents: b⁹ / b⁶ = b^9-6 = b³. This simplifies the equation and makes it much easier to solve for b. Once we have found the value of b, we can then substitute it back into either of the original equations to solve for a. This method highlights the interconnectedness of the parameters in the exponential function and how understanding their relationships can lead to efficient problem-solving strategies. It's important to choose the most appropriate method for solving a system of equations, and in this case, elimination by division proves to be a particularly effective technique. By carefully manipulating the equations and applying the rules of exponents, we can systematically unravel the unknowns and reveal the underlying structure of the exponential function.

The Great Divide: Isolating 'b'

Let's divide equation (2) by equation (1):

(704) / (11) = (ab⁹) / (ab⁶)

Simplifying this gives us:

64 = b³

Now we're cooking! We have a simple equation to solve for b. What number, when cubed, equals 64? The answer, my friends, is 4! So, b = 4. Woohoo! We've found one piece of the puzzle. The cube root of 64 is a fundamental mathematical concept, and recognizing it quickly can save valuable time in problem-solving. The process of taking the cube root is the inverse operation of cubing a number, and it essentially undoes the exponentiation. In this case, finding the cube root of 64 allows us to isolate b and determine its value. This step is crucial because b is the base of the exponential function, and it dictates the rate of growth. Knowing the value of b allows us to understand how the function is changing as x increases. The simplicity of this equation highlights the elegance of the division method we employed earlier. By carefully choosing our strategy, we were able to transform a complex system of equations into a single, easily solvable equation for b. This demonstrates the power of mathematical techniques and the importance of selecting the most efficient approach for a given problem.

Finding 'a': Back to the Basics

Now that we know b = 4, we can plug this value back into either equation (1) or (2) to solve for a. Let's use equation (1) because it looks a little simpler:

11 = a(4)⁶

11 = a * 4096

To isolate a, we divide both sides by 4096:

a = 11 / 4096

We've found a! It's a bit of a fraction, but that's perfectly fine. Now we know both a and b, meaning we have the complete recipe for our function. The process of substituting the value of b back into one of the original equations is a standard technique in solving systems of equations. It allows us to leverage the information we have already obtained to find the remaining unknowns. In this case, substituting b = 4 into the equation 11 = ab⁶ allows us to isolate a and solve for its value. The arithmetic involved in calculating 4⁶ and then dividing 11 by the result might seem daunting, but it's a crucial step in determining the precise value of a. The fraction 11/4096 might appear unusual, but it's a perfectly valid value for a, and it represents the vertical stretch or compression factor of the exponential function. This value is essential for accurately predicting the behavior of the function and for calculating g(11). By systematically working through the steps and carefully applying algebraic techniques, we have successfully determined the values of both a and b, which completes our understanding of the exponential function.

The Grand Finale: Calculating g(11)

We now know that our function is:

g(x) = (11/4096)(4)^x

To find g(11), we simply plug in x = 11:

g(11) = (11/4096)(4)¹¹

g(11) = (11/4096) * 4194304

g(11) = 11264

And there you have it! The value of g(11) is 11264. We've successfully navigated the exponential landscape and emerged victorious! The final step of calculating g(11) is the culmination of all our previous efforts. By plugging in x = 11 into the complete function, we are essentially using the exponential model we have built to predict the output at a specific input value. The arithmetic involved in calculating 4¹¹ and then multiplying by 11/4096 might seem intimidating, but it's a straightforward application of the function's formula. The result, g(11) = 11264, is the answer we have been seeking, and it represents the value of the exponential function at x = 11. This process highlights the power of mathematical modeling and how we can use functions to represent real-world phenomena and make predictions about their behavior. By systematically working through the problem, applying algebraic techniques, and understanding the properties of exponential functions, we have successfully calculated g(11) and demonstrated our mastery of this mathematical concept. This journey through exponential functions has highlighted the power of careful analysis, strategic problem-solving, and a dash of algebraic wizardry. Remember, guys, math is an adventure, and every problem is a chance to learn something new!

Conclusion

In summary, by utilizing the given information g(6) = 11 and g(9) = 704, we were able to determine the values of a and b in the exponential function g(x) = ab^x. This allowed us to precisely define the function as g(x) = (11/4096)(4)^x and subsequently calculate g(11) to be 11264. This problem showcases the elegance and power of exponential functions in modeling relationships where growth or decay occurs at a rate proportional to the current value. The techniques employed, such as solving systems of equations and manipulating exponents, are fundamental tools in mathematical problem-solving. Remember to always break down complex problems into smaller, manageable steps and to leverage the properties of mathematical concepts to simplify your calculations. Keep exploring, keep learning, and keep having fun with math! This is just one small peak in the vast landscape of mathematical knowledge, and there are countless more exciting discoveries waiting to be made.