Evaluating $f(1)$ For $f(x) = 4(\frac{1}{3})^{2x}$

Understanding the Function $f(x) = 4\left(\frac{1}{3}\right)^{2 x}$

Let's dive into the world of functions with our expression, $f(x) = 4\left(\frac{1}{3}\right)^{2 x}$. In mathematics, a function is like a machine that takes an input, does something to it, and gives you an output. Here, our function $f(x)$ is defined by a specific rule involving exponents. The input is represented by '$x$', and the output is what we get after applying the rule. The expression $4\left(\frac{1}{3}\right)^{2 x}$ tells us to take the number $\frac{1}{3}$, raise it to the power of $2x$, and then multiply the result by 4. Understanding how exponents work is key to solving problems like this. Remember, an exponent tells you how many times to multiply a base number by itself. In this case, the base is $\frac{1}{3}$ and the exponent is $2x$. This means we're looking at a form of exponential decay, as the base $\frac{1}{3}$ is a fraction between 0 and 1. Exponential functions are fundamental in many areas of science, finance, and engineering, describing processes like population growth, radioactive decay, and compound interest. The structure of this particular function, with a constant multiplier (4) and an exponential term involving $x$, is quite common. The factor of 4 at the beginning simply scales the entire exponential curve up or down. The term $2x$ in the exponent modifies how quickly the function changes. If it were just $x$, the decay would be at a certain rate. With $2x$, the decay happens twice as fast. This is because raising a number to the power of $2x$ is the same as raising it to the power of $x$ twice: $\left(\frac{1}{3}\right)^{2x} = \left(\left(\frac{1}{3}\right)^x\right)^2$. Grasping these properties will make evaluating the function at specific points much more straightforward.

Calculating $f(1)$: Plugging in the Value

Now that we've got a handle on our function, $f(x) = 4\left(\frac{1}{3}\right)^{2 x}$, the next logical step is to figure out what happens when we feed it a specific input. The question asks us to find $f(1)$, which means we need to substitute the value $1$ for every occurrence of '$x$' in our function's formula. This is a fundamental concept in function evaluation. We're not changing the function itself; we're just looking at its behavior at a particular point on the graph. So, let's replace '$x$' with '$1$': $f(1) = 4\left(\frac{1}{3}\right)^{2 \times 1}$. The first thing to simplify is the exponent. $2 \times 1$ is simply $2$. So, our expression becomes $f(1) = 4\left(\frac{1}{3}\right)^2$. Now, we need to deal with the exponentiation. $\left(\frac{1}{3}\right)^2$ means we multiply $\frac{1}{3}$ by itself: $\frac{1}{3} \times \frac{1}{3}$. When multiplying fractions, you multiply the numerators together and the denominators together. So, $1 \times 1 = 1$ for the numerator, and $3 \times 3 = 9$ for the denominator. This gives us $\frac{1}{9}$. Now, we substitute this back into our equation for $f(1)$: $f(1) = 4 \times \frac{1}{9}$. Finally, we perform the multiplication. Multiplying a whole number by a fraction is as simple as multiplying the whole number by the numerator of the fraction and keeping the same denominator. So, $4 \times 1 = 4$, and the denominator remains $9$. This leads us to the final result: $f(1) = \frac{4}{9}$. This step-by-step process of substitution and simplification is the core of evaluating functions. It's important to follow the order of operations (PEMDAS/BODMAS) – Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) – to ensure accuracy. In this case, we first dealt with the multiplication within the exponent, then the exponentiation itself, and finally the multiplication by the constant factor.

Comparing the Result with the Options

We've successfully calculated that $f(1) = \frac{4}{9}$ by carefully substituting $1$ for $x$ in the function $f(x) = 4\left(\frac{1}{3}\right)^{2 x}$ and following the rules of exponents and fraction multiplication. Now comes the final step: matching our answer to the provided options. The options are:

A. $\frac{4}{3}$ B. $\frac{1}{9}$ C. $\frac{4}{9}$ D. $\frac{4}{27}$ E. $\frac{4}{81}$

Let's quickly review our calculation. We had $f(1) = 4\left(\frac{1}{3}\right)^{2 \times 1}$. This simplified to $f(1) = 4\left(\frac{1}{3}\right)^2$. Squaring the fraction $\frac{1}{3}$ gave us $\frac{1}{9}$. Then, multiplying by 4, we got $4 \times \frac{1}{9} = \frac{4}{9}$.

Comparing $\frac{4}{9}$ to the given choices, we can see that it perfectly matches option C. It's always a good idea to double-check your work, especially if the numbers seem a bit off or if you're unsure. For instance, if we had mistakenly calculated $\left(\frac{1}{3}\right)^2$ as $\frac{1}{6}$ (by adding the denominators instead of multiplying), we would have gotten $4 \times \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$, which is not an option. Or, if we had forgotten to multiply by 4, our answer would be $\frac{1}{9}$, which is option B. If we had incorrectly applied the exponent, perhaps thinking $2x$ meant just multiplying the base by 2, we might have ended up with $4 \times \left(\frac{1}{3} \times 2\right) = 4 \times \frac{2}{3} = \frac{8}{3}$, which is not an option. These checks help reinforce our understanding and catch potential errors. Therefore, based on our accurate calculation, the correct answer is indeed $\frac{4}{9}$.

Conclusion

We've successfully navigated the process of evaluating a function at a specific point. By understanding the components of the function $f(x) = 4\left(\frac{1}{3}\right)^{2 x}$ and carefully applying the rules of exponents and arithmetic, we determined that $f(1)$ equals $\frac{4}{9}$. This problem highlights the importance of precise calculation and understanding mathematical notation. Whether you're working with simple algebraic expressions or complex calculus problems, the ability to substitute values and simplify correctly is a foundational skill. Remember to always pay close attention to the order of operations and to double-check your work. For further exploration into the fascinating world of functions and their applications, you can visit reputable mathematical resources such as the Khan Academy or Wolfram MathWorld for in-depth explanations and examples.