What Is \[$(f \cdot G)(x)\$\]?Given:$\[ \begin{array}{l} f(x) = X^4 - 9 \\ g(x) = X^3 + 9 \end{array} \\]Enter Your Answer In Standard Form In The Box. \[$(f \cdot G)(x) = \square\$\]

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\begin{array}{l} f(x) = x^4 - 9 \ g(x) = x^3 + 9 \end{array} }$

Understanding the Concept of Function Composition

In mathematics, function composition is a process of combining two or more functions to create a new function. This new function takes the output of one function as the input for another function. The concept of function composition is denoted by the symbol $\circ$ and is defined as $(f \circ g)(x) = f(g(x))$. However, in this problem, we are asked to find the composition of two functions $f(x)$ and $g(x)$, denoted by $(f \cdot g)(x)$.

The Composition of Two Functions

To find the composition of two functions $f(x)$ and $g(x)$, we need to substitute the expression for $g(x)$ into the expression for $f(x)$. This means that we will replace the variable $x$ in the expression for $f(x)$ with the expression for $g(x)$. In this case, we have:

$${ (f \cdot g)(x) = f(g(x)) }$$

Substituting the expression for $g(x)$ into the expression for $f(x)$, we get:

$${ (f \cdot g)(x) = f(x^3 + 9) }$$

Evaluating the Composition of Two Functions

Now that we have the expression for the composition of two functions, we can evaluate it by substituting the expression for $g(x)$ into the expression for $f(x)$. This means that we will replace the variable $x$ in the expression for $f(x)$ with the expression for $g(x)$. In this case, we have:

$${ (f \cdot g)(x) = (x^3 + 9)^4 - 9 }$$

Simplifying the Expression

To simplify the expression, we can expand the binomial $(x^3 + 9)^4$ using the binomial theorem. The binomial theorem states that for any positive integer $n$, we have:

$${ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k }$$

In this case, we have $a = x^3$, $b = 9$, and $n = 4$. Substituting these values into the binomial theorem, we get:

$${ (x^3 + 9)^4 = \sum_{k=0}^{4} \binom{4}{k} (x^3)^{4-k} 9^k }$$

Evaluating the Binomial Expansion

To evaluate the binomial expansion, we need to calculate the values of the binomial coefficients and the powers of $x^3$ and $9$. The binomial coefficients are given by:

$${ \binom{4}{k} = \frac{4!}{k!(4-k)!} }$$

The powers of $x^3$ and $9$ are given by:

$${ (x^3)^{4-k} = x^{12-3k} }$$

$${ 9^k = 9^k }$$

Simplifying the Binomial Expansion

To simplify the binomial expansion, we can calculate the values of the binomial coefficients and the powers of $x^3$ and $9$. The binomial coefficients are given by:

$${ \binom{4}{0} = 1 }$$

$${ \binom{4}{1} = 4 }$$

$${ \binom{4}{2} = 6 }$$

$${ \binom{4}{3} = 4 }$$

$${ \binom{4}{4} = 1 }$$

The powers of $x^3$ and $9$ are given by:

$${ (x^3)^4 = x^{12} }$$

$${ (x^3)^3 = x^9 }$$

$${ (x^3)^2 = x^6 }$$

$${ (x^3)^1 = x^3 }$$

$${ (x^3)^0 = 1 }$$

$${ 9^4 = 6561 }$$

$${ 9^3 = 729 }$$

$${ 9^2 = 81 }$$

$${ 9^1 = 9 }$$

$${ 9^0 = 1 }$$

Combining the Terms

To combine the terms, we can add the corresponding terms in the binomial expansion. This gives us:

$${ (x^3 + 9)^4 = x^{12} + 4x^9(9) + 6x^6(81) + 4x^3(729) + 6561 }$$

$${ (x^3 + 9)^4 = x^{12} + 36x^9 + 486x^6 + 2916x^3 + 6561 }$$

Substituting the Binomial Expansion into the Expression

Now that we have the binomial expansion, we can substitute it into the expression for the composition of two functions. This gives us:

$${ (f \cdot g)(x) = (x^3 + 9)^4 - 9 }$$

$${ (f \cdot g)(x) = x^{12} + 36x^9 + 486x^6 + 2916x^3 + 6561 - 9 }$$

$${ (f \cdot g)(x) = x^{12} + 36x^9 + 486x^6 + 2916x^3 + 6552 }$$

The final answer is $\boxed{x^{12} + 36x^9 + 486x^6 + 2916x^3 + 6552}$.

Q: What is function composition?

A: Function composition is a process of combining two or more functions to create a new function. This new function takes the output of one function as the input for another function.

Q: How is function composition denoted?

A: Function composition is denoted by the symbol $\circ$ and is defined as $(f \circ g)(x) = f(g(x))$. However, in this problem, we are asked to find the composition of two functions $f(x)$ and $g(x)$, denoted by $(f \cdot g)(x)$.

Q: What is the difference between function composition and function evaluation?

A: Function composition is the process of combining two or more functions to create a new function, while function evaluation is the process of finding the value of a function at a given input.

Q: How do you evaluate the composition of two functions?

A: To evaluate the composition of two functions, you need to substitute the expression for one function into the expression for the other function. This means that you will replace the variable $x$ in the expression for one function with the expression for the other function.

Q: What is the binomial theorem?

A: The binomial theorem is a mathematical formula that describes the expansion of a binomial raised to a power. It states that for any positive integer $n$, we have:

$${ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k }$$

Q: How do you use the binomial theorem to expand a binomial?

A: To use the binomial theorem to expand a binomial, you need to calculate the values of the binomial coefficients and the powers of the variables. The binomial coefficients are given by:

$${ \binom{n}{k} = \frac{n!}{k!(n-k)!} }$$

The powers of the variables are given by:

$${ a^{n-k} = a^{n-k} }$$

$${ b^k = b^k }$$

Q: What is the difference between a binomial coefficient and a power of a variable?

A: A binomial coefficient is a number that represents the number of ways to choose $k$ items from a set of $n$ items, while a power of a variable is a number that represents the variable raised to a certain exponent.

Q: How do you simplify a binomial expansion?

A: To simplify a binomial expansion, you need to combine the corresponding terms in the expansion. This means that you will add the terms that have the same power of the variable.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\boxed{x^{12} + 36x^9 + 486x^6 + 2916x^3 + 6552}$.

Q: What is the purpose of function composition?

A: The purpose of function composition is to create a new function by combining two or more existing functions. This can be useful in a variety of applications, such as solving equations, modeling real-world phenomena, and optimizing systems.

Q: How do you use function composition in real-world applications?

A: Function composition is used in a variety of real-world applications, such as:

• Solving equations: Function composition can be used to solve equations by combining two or more functions to create a new function that represents the solution to the equation.
• Modeling real-world phenomena: Function composition can be used to model real-world phenomena, such as population growth, chemical reactions, and economic systems.
• Optimizing systems: Function composition can be used to optimize systems, such as finding the maximum or minimum value of a function.

Q: What are some common mistakes to avoid when working with function composition?

A: Some common mistakes to avoid when working with function composition include:

• Not evaluating the composition of two functions correctly
• Not simplifying the binomial expansion correctly
• Not combining the corresponding terms in the binomial expansion correctly
• Not using the correct notation for function composition

Q: How do you troubleshoot common mistakes when working with function composition?

A: To troubleshoot common mistakes when working with function composition, you can:

• Check your work carefully to ensure that you have evaluated the composition of two functions correctly
• Simplify the binomial expansion correctly
• Combine the corresponding terms in the binomial expansion correctly
• Use the correct notation for function composition

Q: What are some tips for working with function composition?

A: Some tips for working with function composition include:

• Make sure to evaluate the composition of two functions correctly
• Simplify the binomial expansion correctly
• Combine the corresponding terms in the binomial expansion correctly
• Use the correct notation for function composition
• Check your work carefully to ensure that you have not made any mistakes.