Evaluating Functions Calculating F(3) For F(x)=(1/7)(7^x)

In the realm of mathematics, functions serve as fundamental building blocks, mapping inputs to corresponding outputs. Evaluating functions for specific input values is a crucial skill. This article delves into the process of evaluating the function $f(x) = (\frac{1}{7})(7^x)$ at $x = 3$. We will explore the step-by-step calculation, providing a clear and comprehensive explanation to enhance understanding. Mastery of function evaluation is essential for tackling more complex mathematical concepts, making this exercise an important stepping stone in your mathematical journey. Let's embark on this exploration and uncover the value of $f(3)$. Understanding functions is critical to solve the math, and this article provides a detailed and easy-to-understand explanation of the solution.

The problem at hand presents us with a function, specifically $f(x) = (\frac{1}{7})(7^x)$. Our objective is to determine the value of this function when the input, denoted by $x$, is set to 3. In mathematical notation, this translates to finding the value of $f(3)$. This type of problem falls under the category of function evaluation, a fundamental skill in algebra and calculus. To successfully solve this, we need to substitute the value of $x$ with 3 in the function's expression and then simplify the resulting expression. Solving for $f(3)$ requires careful application of exponent rules and arithmetic operations. Function evaluation is a core concept in mathematics, and mastering it is essential for solving more complex problems. This article guides you through each step to ensure a clear understanding of the solution.

To find the value of $f(3)$, we will follow a step-by-step approach, ensuring clarity and accuracy in our calculations. Here's how we proceed:

1. Substitute $x$ with 3 in the function $f(x) = (\frac{1}{7})(7^x)$. This gives us $f(3) = (\frac{1}{7})(7^3)$. Substituting the value is the first key step in function evaluation. It allows us to transform the function expression into a numerical expression that we can simplify. This process is fundamental to understanding how functions behave and how their outputs change with different inputs. Accurate substitution is crucial for obtaining the correct result. In our case, we are replacing the variable $x$ with the specific value 3, which sets the stage for the subsequent calculations.

2. Evaluate $7^3$. Recall that $7^3$ means $7 * 7 * 7$, which equals 343. Understanding exponents is essential for this step. The exponent 3 indicates that the base, 7, is multiplied by itself three times. This calculation is a straightforward application of the definition of exponents. Understanding exponents is a fundamental concept in mathematics, particularly in algebra and calculus. It is crucial for simplifying expressions and solving equations. In this case, correctly evaluating $7^3$ is a necessary step to find the value of $f(3)$.

3. Multiply the result from the previous step by $\frac{1}{7}$. So, $f(3) = (\frac{1}{7}) * 343$. Multiplying a fraction by a whole number involves multiplying the numerator of the fraction by the whole number. In this case, we multiply 1 by 343, which gives us 343. Then, we divide the result by the denominator, 7. Performing this multiplication gives us $f(3) = \frac{343}{7}$. This step combines the result of the exponentiation with the fractional coefficient in the function's expression, bringing us closer to the final answer.

4. Simplify the fraction $\frac{343}{7}$. Dividing 343 by 7 gives us 49. Simplification is a critical step in mathematical problem-solving. It involves reducing an expression to its simplest form. In this case, we simplify a fraction by dividing both the numerator and the denominator by their greatest common divisor. Simplifying fractions often makes the answer clearer and easier to interpret. In our problem, simplifying $\frac{343}{7}$ to 49 provides the final numerical value of the function at $x=3$.

Therefore, $f(3) = 49$.

Let's delve deeper into the calculations to ensure a solid understanding. The function we are given is $f(x) = (\frac{1}{7})(7^x)$. Our goal is to find $f(3)$, which means we need to substitute $x$ with 3. This gives us $f(3) = (\frac{1}{7})(7^3)$.

The next step is to evaluate $7^3$. This means we need to calculate 7 raised to the power of 3, which is $7 * 7 * 7$. Multiplying the first two 7s, we get $7 * 7 = 49$. Then, multiplying this result by the remaining 7, we have $49 * 7 = 343$. So, $7^3 = 343$. Understanding the order of operations is crucial here, where exponentiation takes precedence before multiplication.

Now we substitute $7^3$ with 343 in our expression for $f(3)$. This gives us $f(3) = (\frac{1}{7})(343)$. To multiply a fraction by a whole number, we multiply the numerator of the fraction by the whole number and keep the same denominator. In this case, we multiply 1 by 343, which gives us 343. So, we have $f(3) = \frac{343}{7}$. Fraction multiplication is a fundamental arithmetic operation that is essential in various mathematical contexts.

Finally, we simplify the fraction $\frac{343}{7}$. We need to divide 343 by 7. Performing this division, we find that $343 ÷ 7 = 49$. Therefore, $f(3) = 49$. Division and simplification are the final steps in finding the value of the function. They ensure that the result is expressed in its simplest and most understandable form. In this case, dividing 343 by 7 gives us the integer 49, which is the final answer.

When evaluating functions, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your accuracy. One common mistake is misunderstanding the order of operations. For example, some students might mistakenly multiply $\frac{1}{7}$ by 7 first and then raise the result to the power of 3. However, according to the order of operations (PEMDAS/BODMAS), exponentiation should be performed before multiplication. Following the correct order of operations is crucial for obtaining the correct answer.

Another mistake is making errors in the calculation of exponents. For instance, a student might incorrectly calculate $7^3$ as $7 * 3 = 21$ instead of $7 * 7 * 7 = 343$. To avoid this, it's essential to understand the definition of exponents and practice exponentiation. Accurate calculation of exponents is a fundamental skill that is essential for function evaluation and other mathematical operations.

A third common mistake is making errors in fraction multiplication and simplification. For example, a student might correctly calculate $((\frac{1}{7})(343)$ but then make an error when dividing 343 by 7. To avoid this, it's important to have a solid understanding of fraction arithmetic and practice simplifying fractions. Mastering fraction operations is a key component of mathematical proficiency.

Finally, some students might struggle with the concept of function notation and make mistakes when substituting values into the function. For instance, they might substitute $x$ with 3 in the wrong place or not substitute it at all. To avoid this, it's crucial to understand what function notation means and to carefully substitute values into the function expression. Understanding function notation is a core concept in algebra and calculus.

In conclusion, we have successfully evaluated the function $f(x) = (\frac{1}{7})(7^x)$ at $x = 3$ and found that $f(3) = 49$. We achieved this by following a step-by-step approach that involved substituting the value of $x$, evaluating the exponent, multiplying the result by the fraction, and simplifying. Throughout the solution, we emphasized the importance of understanding the order of operations, accurately calculating exponents, mastering fraction operations, and understanding function notation. Mastering these skills is essential for success in mathematics.

By avoiding common mistakes and practicing function evaluation, you can build confidence in your mathematical abilities. Function evaluation is a fundamental skill that is used in many areas of mathematics, including algebra, calculus, and differential equations. Therefore, a solid understanding of this concept is crucial for further mathematical studies. Continued practice and a thorough understanding of the underlying principles will solidify your ability to evaluate functions accurately and efficiently.