Given The Functions F ( X ) = 6 X 3 + 7 F(x)=6x^3+7 F ( X ) = 6 X 3 + 7 And G ( X ) = 3 X G(x)=3^x G ( X ) = 3 X , Evaluate F ( G ( − 2 ) F(g(-2) F ( G ( − 2 ) ].

Introduction

In mathematics, composite functions are a fundamental concept that allows us to combine two or more functions to create a new function. Evaluating composite functions requires a clear understanding of the individual functions and their properties. In this article, we will evaluate the composite function $f(g(-2))$ using the given functions $f(x)=6x^3+7$ and $g(x)=3^x$.

Understanding the Given Functions

Before we proceed with evaluating the composite function, let's take a closer look at the given functions.

Function $f(x)$

The function $f(x)$ is defined as $f(x)=6x^3+7$. This is a polynomial function of degree 3, where the coefficient of the cubic term is 6 and the constant term is 7.

Function $g(x)$

The function $g(x)$ is defined as $g(x)=3^x$. This is an exponential function with base 3.

Evaluating the Composite Function

Now that we have a clear understanding of the given functions, let's proceed with evaluating the composite function $f(g(-2))$.

Step 1: Evaluate $g(-2)$

To evaluate the composite function, we need to start by evaluating the inner function $g(-2)$. Substituting $x=-2$ into the function $g(x)=3^x$, we get:

$$g(-2)=3^{-2}=\frac{1}{3^2}=\frac{1}{9}$$

Step 2: Evaluate $f(g(-2))$

Now that we have evaluated the inner function $g(-2)$, we can proceed with evaluating the outer function $f(x)=6x^3+7$. Substituting $x=\frac{1}{9}$ into the function $f(x)$, we get:

$$f(g(-2))=f\left(\frac{1}{9}\right)=6\left(\frac{1}{9}\right)^3+7$$

Simplifying the Expression

To simplify the expression, we can start by evaluating the cubic term:

$$\left(\frac{1}{9}\right)^3=\frac{1}{729}$$

Now, we can substitute this value back into the expression:

$$f(g(-2))=6\left(\frac{1}{729}\right)+7$$

Final Evaluation

To evaluate the expression, we can start by multiplying the coefficient 6 by the fraction $\frac{1}{729}$:

$$6\left(\frac{1}{729}\right)=\frac{6}{729}$$

Now, we can add 7 to this value:

$$f(g(-2))=\frac{6}{729}+7$$

To add these two values, we need to find a common denominator. The least common multiple of 729 and 1 is 729, so we can rewrite 7 as:

$$7=\frac{7\cdot729}{729}=\frac{5103}{729}$$

Now, we can add the two values:

$$f(g(-2))=\frac{6}{729}+\frac{5103}{729}=\frac{5109}{729}$$

Conclusion

In this article, we evaluated the composite function $f(g(-2))$ using the given functions $f(x)=6x^3+7$ and $g(x)=3^x$. We started by evaluating the inner function $g(-2)$, and then proceeded with evaluating the outer function $f(x)$. By simplifying the expression and finding a common denominator, we were able to evaluate the composite function and obtain a final answer.

Final Answer

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Introduction

In our previous article, we evaluated the composite function $f(g(-2))$ using the given functions $f(x)=6x^3+7$ and $g(x)=3^x$. In this article, we will provide a Q&A guide to help you better understand the concept of composite functions and how to evaluate them.

Q&A

Q: What is a composite function?

A: A composite function is a function that is formed by combining two or more functions. It is a way of creating a new function by using the output of one function as the input for another function.

Q: How do I evaluate a composite function?

A: To evaluate a composite function, you need to start by evaluating the inner function. Once you have evaluated the inner function, you can use its output as the input for the outer function. You can then evaluate the outer function using the input from the inner function.

Q: What is the order of operations for evaluating a composite function?

A: The order of operations for evaluating a composite function is as follows:

1. Evaluate the inner function.
2. Use the output of the inner function as the input for the outer function.
3. Evaluate the outer function using the input from the inner function.

Q: How do I simplify an expression when evaluating a composite function?

A: When simplifying an expression when evaluating a composite function, you can start by evaluating any exponential terms. You can then multiply any coefficients by the terms. Finally, you can add or subtract any like terms.

Q: What is the difference between a composite function and a function of a function?

A: A composite function is a function that is formed by combining two or more functions. A function of a function is a function that takes another function as its input. While both concepts involve functions, they are not the same thing.

Q: Can I use a composite function to solve a problem in real life?

A: Yes, composite functions can be used to solve a variety of problems in real life. For example, you can use a composite function to model population growth, financial transactions, or other complex systems.

Q: How do I know when to use a composite function?

A: You should use a composite function when you need to combine two or more functions to solve a problem. This can be the case when you need to model a complex system, or when you need to evaluate a function that is not easily evaluated on its own.

Q: Can I use a composite function with different types of functions?

A: Yes, you can use a composite function with different types of functions. For example, you can use a composite function with a polynomial function and an exponential function.

Q: How do I evaluate a composite function with multiple inner functions?

A: To evaluate a composite function with multiple inner functions, you need to start by evaluating the innermost function. Once you have evaluated the innermost function, you can use its output as the input for the next inner function. You can then continue this process until you have evaluated all of the inner functions.

Q: Can I use a composite function to evaluate a function that is not easily evaluated on its own?

A: Yes, you can use a composite function to evaluate a function that is not easily evaluated on its own. This can be the case when you need to evaluate a function that involves complex calculations or when you need to evaluate a function that is not easily expressed in terms of elementary functions.

Conclusion

In this article, we provided a Q&A guide to help you better understand the concept of composite functions and how to evaluate them. We covered a variety of topics, including the order of operations for evaluating a composite function, how to simplify an expression when evaluating a composite function, and how to use a composite function to solve a problem in real life.

Final Tips

• Make sure to evaluate the inner function first when evaluating a composite function.
• Use the output of the inner function as the input for the outer function.
• Simplify the expression by evaluating any exponential terms, multiplying any coefficients by the terms, and adding or subtracting any like terms.
• Use a composite function to solve a problem in real life when you need to combine two or more functions to model a complex system.

Final Answer

The final answer is $\boxed{\frac{5109}{729}}$.