Evaluate The Function Rule For The Given Value: $ Y = 6 \cdot 4^x }$For { X = -3 $}$, Choose The Correct Answer A. { -72$ $B. { \frac{3}{128}$}$C. { \frac{3}{8}$}$D. { \frac{3}{32}$}$

Introduction

In mathematics, function rules are used to describe the relationship between input values and output values. Evaluating a function rule for a given value involves substituting the value into the function and simplifying the expression to obtain the corresponding output value. In this article, we will evaluate the function rule $y = 6 \cdot 4^x$ for the given value $x = -3$ and choose the correct answer from the options provided.

Understanding the Function Rule

The function rule $y = 6 \cdot 4^x$ describes a relationship between the input value $x$ and the output value $y$. The function is a product of two terms: a constant term $6$ and an exponential term $4^x$. The exponential term $4^x$ represents the power of $4$ raised to the value of $x$. To evaluate the function rule for a given value, we need to substitute the value into the function and simplify the expression.

Evaluating the Function Rule for x = -3

To evaluate the function rule for $x = -3$, we substitute the value into the function and simplify the expression:

$$y = 6 \cdot 4^{-3}$$

Using the property of exponents that states $a^{-n} = \frac{1}{a^n}$, we can rewrite the expression as:

$$y = 6 \cdot \frac{1}{4^3}$$

Simplifying the expression further, we get:

$$y = 6 \cdot \frac{1}{64}$$

Multiplying the numerator and denominator, we get:

$$y = \frac{6}{64}$$

Reducing the fraction to its simplest form, we get:

$$y = \frac{3}{32}$$

Choosing the Correct Answer

Based on the evaluation of the function rule for $x = -3$, we can choose the correct answer from the options provided. The correct answer is:

• D. $\frac{3}{32}$

The other options are incorrect because they do not match the result of the evaluation.

Conclusion

Evaluating a function rule for a given value involves substituting the value into the function and simplifying the expression to obtain the corresponding output value. In this article, we evaluated the function rule $y = 6 \cdot 4^x$ for the given value $x = -3$ and chose the correct answer from the options provided. The correct answer is $\frac{3}{32}$.

Additional Examples

To further illustrate the concept of evaluating a function rule for a given value, let's consider a few additional examples.

Example 1

Evaluate the function rule $y = 2 \cdot 3^x$ for $x = 2$.

Solution

Substituting the value into the function, we get:

$$y = 2 \cdot 3^2$$

Simplifying the expression, we get:

$$y = 2 \cdot 9$$

Multiplying the numerator and denominator, we get:

$$y = 18$$

Example 2

Evaluate the function rule $y = 5 \cdot 2^x$ for $x = -2$.

Solution

Substituting the value into the function, we get:

$$y = 5 \cdot 2^{-2}$$

Using the property of exponents that states $a^{-n} = \frac{1}{a^n}$, we can rewrite the expression as:

$$y = 5 \cdot \frac{1}{2^2}$$

Simplifying the expression further, we get:

$$y = 5 \cdot \frac{1}{4}$$

Multiplying the numerator and denominator, we get:

$$y = \frac{5}{4}$$

Example 3

Evaluate the function rule $y = 4 \cdot 5^x$ for $x = 1$.

Solution

Substituting the value into the function, we get:

$$y = 4 \cdot 5^1$$

Simplifying the expression, we get:

$$y = 4 \cdot 5$$

Multiplying the numerator and denominator, we get:

$$y = 20$$

These examples illustrate the concept of evaluating a function rule for a given value and demonstrate how to apply the concept to different function rules and values.

Conclusion

$$$$$$

Introduction

Evaluating function rules is a fundamental concept in mathematics that involves substituting values into a function and simplifying the expression to obtain the corresponding output value. In our previous article, we evaluated the function rule $y = 6 \cdot 4^x$ for the given value $x = -3$ and chose the correct answer from the options provided. In this article, we will provide a Q&A guide to help you better understand the concept of evaluating function rules.

Q: What is a function rule?

A: A function rule is a mathematical expression that describes the relationship between input values and output values. It is a way to represent a function in a concise and compact form.

Q: How do I evaluate a function rule for a given value?

A: To evaluate a function rule for a given value, you need to substitute the value into the function and simplify the expression to obtain the corresponding output value.

Q: What are some common properties of exponents that I should know?

A: Some common properties of exponents that you should know include:

• $a^m \cdot a^n = a^{m+n}$
• $\frac{a^m}{a^n} = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $a^{-n} = \frac{1}{a^n}$

Q: How do I simplify expressions involving exponents?

A: To simplify expressions involving exponents, you can use the properties of exponents to combine like terms and simplify the expression.

Q: What are some common mistakes to avoid when evaluating function rules?

A: Some common mistakes to avoid when evaluating function rules include:

• Not substituting the value into the function correctly
• Not simplifying the expression correctly
• Not using the properties of exponents correctly

Q: How do I choose the correct answer from the options provided?

A: To choose the correct answer from the options provided, you need to evaluate the function rule for the given value and compare the result to the options provided.

Q: What are some real-world applications of evaluating function rules?

A: Some real-world applications of evaluating function rules include:

• Modeling population growth and decline
• Modeling the spread of diseases
• Modeling the behavior of physical systems
• Modeling the behavior of economic systems

Q: How can I practice evaluating function rules?

A: You can practice evaluating function rules by working through examples and exercises in your textbook or online resources. You can also try creating your own examples and exercises to practice evaluating function rules.

Conclusion

Evaluating function rules is a fundamental concept in mathematics that involves substituting values into a function and simplifying the expression to obtain the corresponding output value. In this article, we provided a Q&A guide to help you better understand the concept of evaluating function rules. We also discussed some common properties of exponents, how to simplify expressions involving exponents, and some common mistakes to avoid when evaluating function rules. By practicing evaluating function rules, you can develop your problem-solving skills and apply the concept to real-world applications.

Additional Resources

For more information on evaluating function rules, you can check out the following resources:

• Khan Academy: Evaluating Functions
• Mathway: Evaluating Functions
• Wolfram Alpha: Evaluating Functions

Conclusion

Evaluating function rules is a critical concept in mathematics that involves substituting values into a function and simplifying the expression to obtain the corresponding output value. By understanding the concept of evaluating function rules, you can develop your problem-solving skills and apply the concept to real-world applications. We hope this Q&A guide has been helpful in understanding the concept of evaluating function rules.