Evaluating Exponential Functions Find F(-1) For F(x) = 6(5)^x

Introduction to Exponential Functions

In the realm of mathematics, exponential functions hold a significant position, describing phenomena that exhibit rapid growth or decay. These functions are characterized by a constant base raised to a variable exponent, taking the general form $f(x) = a(b)^x$, where $a$ is the initial value, $b$ is the base, and $x$ is the exponent. Understanding exponential functions is crucial in various fields, including finance, biology, and physics, as they model processes such as compound interest, population growth, and radioactive decay. The function $f(x) = 6(5)^x$ is a prime example of an exponential function, where the initial value is 6 and the base is 5. This function represents a quantity that grows exponentially as the value of $x$ increases. To delve deeper into the behavior of this function, we can evaluate it at specific values of $x$. One such value is $x = -1$, which allows us to explore the function's behavior when the exponent is negative. Evaluating exponential functions at negative exponents provides insights into their decay behavior and helps us understand how the function changes as the input decreases. In the following sections, we will explore the evaluation of $f(x) = 6(5)^x$ at $x = -1$, demonstrating the step-by-step process and highlighting the underlying mathematical principles. This exploration will not only enhance our understanding of exponential functions but also provide a foundation for tackling more complex problems involving these powerful mathematical tools. The ability to manipulate exponential expressions and solve equations involving them is a fundamental skill in mathematics and its applications. By mastering these concepts, we can unlock the potential of exponential functions to model and analyze real-world phenomena.

Evaluating $f(x) = 6(5)^x$ at $x = -1$

To find the value of the function $f(x) = 6(5)^x$ at $x = -1$, we need to substitute $-1$ for $x$ in the function's expression. This process involves replacing the variable $x$ with the specific value we are interested in, which in this case is $-1$. The substitution gives us $f(-1) = 6(5)^{-1}$. Now, we need to simplify the expression $6(5)^{-1}$. Recall that a negative exponent indicates the reciprocal of the base raised to the positive exponent. In other words, a^{-n} = rac{1}{a^n}. Applying this rule to our expression, we have 5^{-1} = rac{1}{5^1} = rac{1}{5}. Substituting this back into the expression for $f(-1)$, we get f(-1) = 6ig(rac{1}{5}ig). To simplify further, we multiply the fraction by the constant 6, which gives us f(-1) = rac{6}{5}. This is the simplified exact value of the function at $x = -1$. The result, rac{6}{5}, is a fraction, which is a common form for representing exact values in mathematics. It is important to maintain exact values whenever possible, as decimal approximations can introduce rounding errors. Therefore, the exact value of $f(-1)$ for the function $f(x) = 6(5)^x$ is rac{6}{5}. This calculation demonstrates the fundamental process of evaluating exponential functions at specific points, which is a crucial skill in various mathematical and scientific contexts. Understanding how to manipulate exponents and simplify expressions is essential for working with exponential functions and their applications.

Step-by-Step Solution

Let's break down the solution into a clear, step-by-step process for better understanding. This will help in grasping the underlying concepts and applying them to similar problems.

Step 1: Substitute $x = -1$ into the function.

We begin by replacing the variable $x$ in the function $f(x) = 6(5)^x$ with the value $-1$. This substitution gives us: $f(-1) = 6(5)^{-1}$. This is a direct application of the function's definition, where we are simply evaluating the function at a specific input value.

Step 2: Apply the negative exponent rule.

Recall that a negative exponent means taking the reciprocal of the base raised to the positive exponent. The rule states that a^{-n} = rac{1}{a^n}. Applying this rule to the term $5^{-1}$, we get: 5^{-1} = rac{1}{5^1} = rac{1}{5}. Understanding and applying exponent rules is crucial for simplifying expressions involving powers and roots.

Step 3: Substitute the simplified exponent back into the expression.

Now, we substitute the simplified value of $5^{-1}$ back into the expression for $f(-1)$. This gives us: f(-1) = 6ig(rac{1}{5}ig). This step combines the results of the previous steps to bring us closer to the final answer.

Step 4: Multiply the constant by the fraction.

To simplify the expression further, we multiply the constant 6 by the fraction rac{1}{5}. This is a basic arithmetic operation that involves multiplying a whole number by a fraction: f(-1) = rac{6}{5}. The result is a fraction, which represents the exact value of the function at $x = -1$.

Step 5: State the final answer.

Therefore, the simplified exact value of the function $f(x) = 6(5)^x$ at $x = -1$ is rac{6}{5}. This is the final answer, expressed as a fraction in its simplest form. By following these steps, we have successfully evaluated the exponential function at the given input value.

Understanding Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent. This is a fundamental concept in mathematics that is crucial for simplifying expressions and solving equations involving exponents. The rule for negative exponents can be stated as: a^{-n} = rac{1}{a^n}, where $a$ is any non-zero number and $n$ is any integer. This rule tells us that a term with a negative exponent is equivalent to the reciprocal of the same term with a positive exponent. For example, $5^{-1}$ is the same as rac{1}{5^1}, which simplifies to rac{1}{5}. Similarly, $2^{-3}$ is the same as rac{1}{2^3}, which simplifies to rac{1}{8}. Understanding negative exponents is essential for working with exponential functions, as they often appear in expressions and equations. When evaluating exponential functions at negative values of $x$, we need to apply the negative exponent rule to simplify the expression. This involves taking the reciprocal of the base raised to the positive value of the exponent. For instance, in the function $f(x) = 6(5)^x$, when we evaluate $f(-1)$, we encounter the term $5^{-1}$. Applying the negative exponent rule, we know that 5^{-1} = rac{1}{5}. This allows us to simplify the expression and find the value of the function at $x = -1$. Negative exponents also play a crucial role in scientific notation, where they are used to represent very small numbers. For example, the number 0.001 can be written in scientific notation as $1 imes 10^{-3}$. The negative exponent indicates that we need to move the decimal point three places to the left. In summary, understanding negative exponents is a fundamental skill in mathematics that is essential for simplifying expressions, evaluating functions, and working with scientific notation. By mastering this concept, we can confidently tackle a wide range of problems involving exponents and their applications.

Importance of Exact Values

In mathematics, the concept of exact values is paramount, particularly when dealing with functions and their evaluations. An exact value represents the precise numerical result without any approximation or rounding. This is in contrast to decimal approximations, which can introduce errors due to truncation or rounding. When evaluating functions, especially exponential functions, it is crucial to maintain exact values whenever possible. This ensures that our calculations are accurate and that we avoid the accumulation of errors that can occur with repeated approximations. For instance, in the problem we discussed, the exact value of $f(-1)$ for the function $f(x) = 6(5)^x$ is rac{6}{5}. This is a fraction, which is a precise representation of the value. If we were to convert this fraction to a decimal, we would get 1.2. While 1.2 is a decimal representation of the value, it is important to recognize that it may not always be the most accurate representation, especially if further calculations are involved. For example, if we were to use the decimal approximation 1.2 in subsequent calculations, we might introduce rounding errors that could affect the final result. Maintaining exact values is particularly important in fields such as physics and engineering, where precise calculations are essential for accurate modeling and predictions. In these fields, even small errors can have significant consequences. Therefore, it is a best practice to work with exact values whenever possible and to avoid decimal approximations until the final step of a calculation. By adhering to this principle, we can ensure the accuracy and reliability of our mathematical work. In the context of exponential functions, exact values often involve fractions or radicals, which are precise representations of numbers. By understanding the importance of exact values, we can approach mathematical problems with greater rigor and confidence.

Conclusion

In conclusion, we have successfully evaluated the function $f(x) = 6(5)^x$ at $x = -1$, finding the simplified exact value to be rac{6}{5}. This process involved substituting the value of $x$ into the function, applying the negative exponent rule, and simplifying the resulting expression. We have also emphasized the importance of understanding negative exponents and maintaining exact values in mathematical calculations. Evaluating exponential functions is a fundamental skill in mathematics with applications in various fields, including finance, biology, and physics. By mastering these concepts, we can confidently tackle more complex problems involving exponential functions and their applications. The step-by-step solution provided in this article offers a clear and concise approach to evaluating exponential functions at specific points. By following these steps, we can avoid common errors and ensure accurate results. Furthermore, we have highlighted the significance of using exact values whenever possible, as this minimizes the risk of rounding errors and maintains the integrity of our calculations. Exponential functions are a powerful tool for modeling phenomena that exhibit rapid growth or decay. Understanding their properties and how to evaluate them is essential for anyone working in mathematics, science, or engineering. The concepts discussed in this article provide a solid foundation for further exploration of exponential functions and their applications. By continuing to practice and apply these skills, we can deepen our understanding of this important mathematical topic. The ability to work with exponential functions is a valuable asset in a wide range of disciplines, and the knowledge gained from this article will serve as a strong foundation for future learning and problem-solving.