Substituting the Exponent into the Function

Oct 13, 2025 by magicalceylontours.com 137 views

Evaluating the Function f(x) = (1/3)(6)^x for x = 3

In this article, we will evaluate the function f(x) = (1/3)(6)^x for x = 3. This involves substituting the value of x into the function and simplifying the resulting expression. We will use the properties of exponents and basic algebra to evaluate the function.

The function f(x) = (1/3)(6)^x is an exponential function, where the base is 6 and the exponent is x. The function is defined for all real values of x. To evaluate the function for x = 3, we need to substitute 3 into the function in place of x.

Substituting x = 3 into the Function

To substitute x = 3 into the function, we replace x with 3 in the expression (1/3)(6)^x. This gives us:

f(3) = (1/3)(6)^3

Evaluating the Exponent

The next step is to evaluate the exponent 6^3. This involves multiplying 6 by itself three times:

6^3 = 6 × 6 × 6 = 216

Substituting the Exponent into the Function

Now that we have evaluated the exponent, we can substitute it into the function:

f(3) = (1/3)(216)

Simplifying the Expression

To simplify the expression, we can multiply 216 by 1/3:

f(3) = 216 × (1/3) = 72

Exponents are a fundamental concept in mathematics, and they play a crucial role in evaluating functions like f(x) = (1/3)(6)^x. Here are some key properties of exponents:

Product of Powers: When multiplying two powers with the same base, we add the exponents. For example, 2^3 × 2^4 = 2^(3+4) = 2^7.
Power of a Power: When raising a power to another power, we multiply the exponents. For example, (2³⁾4 = 2^(3×4) = 2^12.
Zero Exponent: Any non-zero number raised to the power of 0 is equal to 1. For example, 2^0 = 1.

Exponents have many real-world applications, including:

Finance: Exponents are used to calculate compound interest and investment returns.
Science: Exponents are used to describe the growth and decay of populations, chemical reactions, and physical systems.
Engineering: Exponents are used to design and optimize systems, such as electrical circuits and mechanical systems.

Here are some tips and tricks for evaluating functions like f(x) = (1/3)(6)^x:

Read the problem carefully: Make sure you understand what the problem is asking for.
Substitute values carefully: Make sure you substitute the correct values into the function.
Evaluate exponents carefully: Make sure you evaluate the exponents correctly.
Simplify expressions carefully: Make sure you simplify the expressions correctly.

In this article, we evaluated the function f(x) = (1/3)(6)^x for x = 3. We substituted x = 3 into the function, evaluated the exponent, and simplified the resulting expression. The final answer is f(3) = 72. We also discussed the properties of exponents, real-world applications, and tips and tricks for evaluating functions like f(x) = (1/3)(6)^x.
Evaluating the Function f(x) = (1/3)(6)^x for x = 3: Q&A

In our previous article, we evaluated the function f(x) = (1/3)(6)^x for x = 3. We substituted x = 3 into the function, evaluated the exponent, and simplified the resulting expression. The final answer is f(3) = 72. In this article, we will answer some common questions related to evaluating the function f(x) = (1/3)(6)^x.

Q: What is the base of the exponential function f(x) = (1/3)(6)^x?

A: The base of the exponential function f(x) = (1/3)(6)^x is 6.

Q: What is the exponent of the exponential function f(x) = (1/3)(6)^x?

A: The exponent of the exponential function f(x) = (1/3)(6)^x is x.

Q: How do I evaluate the exponent 6^3?

A: To evaluate the exponent 6^3, you need to multiply 6 by itself three times: 6 × 6 × 6 = 216.

Q: How do I simplify the expression (1/3)(216)?

A: To simplify the expression (1/3)(216), you need to multiply 216 by 1/3: 216 × (1/3) = 72.

Q: What is the final answer for f(3) = (1/3)(6)^3?

A: The final answer for f(3) = (1/3)(6)^3 is 72.

Q: Can I use a calculator to evaluate the function f(x) = (1/3)(6)^x?

A: Yes, you can use a calculator to evaluate the function f(x) = (1/3)(6)^x. However, make sure you understand the concept of exponents and how to evaluate them correctly.

Q: How do I apply the properties of exponents to evaluate the function f(x) = (1/3)(6)^x?

A: To apply the properties of exponents to evaluate the function f(x) = (1/3)(6)^x, you need to follow these steps:

Substitute x = 3 into the function.
Evaluate the exponent 6^3.
Simplify the expression (1/3)(216).

Q: What are some real-world applications of the function f(x) = (1/3)(6)^x?

A: Some real-world applications of the function f(x) = (1/3)(6)^x include:

Finance: Exponents are used to calculate compound interest and investment returns.
Science: Exponents are used to describe the growth and decay of populations, chemical reactions, and physical systems.
Engineering: Exponents are used to design and optimize systems, such as electrical circuits and mechanical systems.

In this article, we answered some common questions related to evaluating the function f(x) = (1/3)(6)^x. We discussed the base and exponent of the function, how to evaluate the exponent, and how to simplify the expression. We also discussed some real-world applications of the function and provided tips and tricks for evaluating functions like f(x) = (1/3)(6)^x.

Here are some tips and tricks for evaluating functions like f(x) = (1/3)(6)^x:

Read the problem carefully: Make sure you understand what the problem is asking for.
Substitute values carefully: Make sure you substitute the correct values into the function.
Evaluate exponents carefully: Make sure you evaluate the exponents correctly.
Simplify expressions carefully: Make sure you simplify the expressions correctly.
Use a calculator wisely: Make sure you understand the concept of exponents and how to evaluate them correctly before using a calculator.

In conclusion, evaluating the function f(x) = (1/3)(6)^x for x = 3 involves substituting x = 3 into the function, evaluating the exponent, and simplifying the resulting expression. The final answer is f(3) = 72. We hope this article has provided you with a better understanding of how to evaluate functions like f(x) = (1/3)(6)^x.