Find $f(x$\] When $x = \left(\frac{1}{3}\right$\] For The Function $y = F(x) = \left(\frac{1}{8}\right)^x$.Round Your Answer To The Nearest Thousandth.

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Introduction

In this article, we will explore how to find the value of a function f(x)f(x) when the input xx is given. We will use a specific function y=f(x)=(18)xy = f(x) = \left(\frac{1}{8}\right)^x and find the value of f(x)f(x) when x=(13)x = \left(\frac{1}{3}\right). This problem involves understanding the concept of exponential functions and how to evaluate them for specific inputs.

Understanding Exponential Functions

Exponential functions are a type of mathematical function that describes a relationship between two quantities. In this case, the function y=f(x)=(18)xy = f(x) = \left(\frac{1}{8}\right)^x describes a relationship between the input xx and the output yy. The base of the exponential function is 18\frac{1}{8}, and the exponent is xx.

Evaluating the Function for a Specific Input

To find the value of f(x)f(x) when x=(13)x = \left(\frac{1}{3}\right), we need to substitute xx into the function and evaluate it. This can be done using the following steps:

  1. Substitute x=(13)x = \left(\frac{1}{3}\right) into the function y=f(x)=(18)xy = f(x) = \left(\frac{1}{8}\right)^x.
  2. Evaluate the expression (18)13\left(\frac{1}{8}\right)^{\frac{1}{3}}.

Evaluating the Expression

To evaluate the expression (18)13\left(\frac{1}{8}\right)^{\frac{1}{3}}, we can use the following steps:

  1. Rewrite the expression as (123)13\left(\frac{1}{2^3}\right)^{\frac{1}{3}}.
  2. Use the property of exponents that states (am)n=amn\left(a^m\right)^n = a^{mn} to simplify the expression.
  3. Evaluate the expression (12)1\left(\frac{1}{2}\right)^1.

Simplifying the Expression

Using the property of exponents, we can simplify the expression as follows:

(123)13=(12)13â‹…3=(12)1=12\left(\frac{1}{2^3}\right)^{\frac{1}{3}} = \left(\frac{1}{2}\right)^{\frac{1}{3} \cdot 3} = \left(\frac{1}{2}\right)^1 = \frac{1}{2}

Rounding the Answer to the Nearest Thousandth

The value of f(x)f(x) when x=(13)x = \left(\frac{1}{3}\right) is 12\frac{1}{2}. To round this answer to the nearest thousandth, we need to evaluate the expression 12\frac{1}{2} to three decimal places.

12=0.500\frac{1}{2} = 0.500

Conclusion

In this article, we found the value of f(x)f(x) when x=(13)x = \left(\frac{1}{3}\right) for the function y=f(x)=(18)xy = f(x) = \left(\frac{1}{8}\right)^x. We used the concept of exponential functions and evaluated the expression (18)13\left(\frac{1}{8}\right)^{\frac{1}{3}} to find the value of f(x)f(x). Finally, we rounded the answer to the nearest thousandth.

Final Answer

The final answer is 0.500\boxed{0.500}.

Related Topics

  • Exponential functions
  • Evaluating expressions
  • Rounding numbers

References

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Introduction

In our previous article, we explored how to find the value of a function f(x)f(x) when the input xx is given. We used a specific function y=f(x)=(18)xy = f(x) = \left(\frac{1}{8}\right)^x and found the value of f(x)f(x) when x=(13)x = \left(\frac{1}{3}\right). In this article, we will answer some frequently asked questions (FAQs) about finding the value of f(x)f(x).

Q&A

Q: What is the formula for finding the value of f(x)f(x)?

A: The formula for finding the value of f(x)f(x) is y=f(x)=axy = f(x) = a^x, where aa is the base and xx is the exponent.

Q: How do I evaluate an exponential function?

A: To evaluate an exponential function, you need to substitute the value of xx into the function and evaluate the expression.

Q: What is the difference between an exponential function and a polynomial function?

A: An exponential function is a function that describes a relationship between two quantities where one quantity is raised to a power of the other quantity. A polynomial function is a function that describes a relationship between two quantities where the variables are added, subtracted, multiplied, or divided.

Q: Can I use a calculator to find the value of f(x)f(x)?

A: Yes, you can use a calculator to find the value of f(x)f(x). However, make sure to use the correct formula and follow the correct steps to evaluate the expression.

Q: How do I round the answer to the nearest thousandth?

A: To round the answer to the nearest thousandth, you need to evaluate the expression to three decimal places.

Q: What is the significance of the base in an exponential function?

A: The base in an exponential function determines the rate at which the function grows or decays. A base greater than 1 will result in a function that grows exponentially, while a base less than 1 will result in a function that decays exponentially.

Q: Can I use a graphing calculator to visualize the exponential function?

A: Yes, you can use a graphing calculator to visualize the exponential function. This can help you understand the behavior of the function and make it easier to evaluate the expression.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about finding the value of f(x)f(x). We covered topics such as the formula for finding the value of f(x)f(x), evaluating exponential functions, and rounding the answer to the nearest thousandth. We also discussed the significance of the base in an exponential function and how to use a graphing calculator to visualize the function.

Related Topics

  • Exponential functions
  • Evaluating expressions
  • Rounding numbers
  • Graphing calculators

References