If F ( X ) = X 3 − 1 F(x)=x^3-1 F ( X ) = X 3 − 1 And G ( X ) = ( X + 6 ) 2 G(x)=(x+6)^2 G ( X ) = ( X + 6 ) 2 , Find ( F ⋅ G ) ( − 4 (f \cdot G)(-4 ( F ⋅ G ) ( − 4 ].

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Introduction

In mathematics, the concept of function composition is a fundamental idea that allows us to combine two or more functions to create a new function. The composition of two functions, $f(x)$ and $g(x)$, is denoted by $(f \cdot g)(x)$ and is defined as $(f \cdot g)(x) = f(g(x))$. In this article, we will explore the composition of two given functions, $f(x)=x^3-1$ and $g(x)=(x+6)^2$, and find the value of $(f \cdot g)(-4)$.

Understanding the Functions

Before we proceed with the composition, let's take a closer look at the two given functions.

Function $f(x)$

The function $f(x)$ is defined as $f(x) = x^3 - 1$. This is a cubic function, which means it has a degree of 3. The graph of this function is a cubic curve that opens upwards.

Function $g(x)$

The function $g(x)$ is defined as $g(x) = (x+6)^2$. This is a quadratic function, which means it has a degree of 2. The graph of this function is a parabola that opens upwards.

Composition of Functions

Now that we have a good understanding of the two functions, let's proceed with the composition. The composition of $f(x)$ and $g(x)$ is denoted by $(f \cdot g)(x)$ and is defined as $(f \cdot g)(x) = f(g(x))$. In other words, we need to plug in the expression for $g(x)$ into the function $f(x)$.

Finding $(f \cdot g)(x)$

To find the composition $(f \cdot g)(x)$, we need to substitute the expression for $g(x)$ into the function $f(x)$. This gives us:

$(f \cdot g)(x) = f(g(x)) = f((x+6)^2) = ((x+6)^2)^3 - 1$

Simplifying this expression, we get:

$(f \cdot g)(x) = (x+6)^6 - 1$

Finding $(f \cdot g)(-4)$

Now that we have the composition $(f \cdot g)(x)$, we can find the value of $(f \cdot g)(-4)$ by plugging in $x = -4$ into the expression.

$(f \cdot g)(-4) = ( (-4+6)^6 ) - 1$

Simplifying this expression, we get:

$(f \cdot g)(-4) = (2^6) - 1$

$(f \cdot g)(-4) = 64 - 1$

$(f \cdot g)(-4) = 63$

Conclusion

In this article, we explored the composition of two given functions, $f(x)=x^3-1$ and $g(x)=(x+6)^2$, and found the value of $(f \cdot g)(-4)$. We started by understanding the two functions, then proceeded with the composition, and finally found the value of $(f \cdot g)(-4)$ by plugging in $x = -4$ into the expression. The final answer is 63.

Final Answer

The final answer is $\boxed{63}$.

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Introduction

In our previous article, we explored the composition of two given functions, $f(x)=x^3-1$ and $g(x)=(x+6)^2$, and found the value of $(f \cdot g)(-4)$. In this article, we will answer some frequently asked questions related to the composition of functions and provide additional insights.

Q&A

Q1: What is the composition of functions?

A1: The composition of two functions, $f(x)$ and $g(x)$, is denoted by $(f \cdot g)(x)$ and is defined as $(f \cdot g)(x) = f(g(x))$. This means that we need to plug in the expression for $g(x)$ into the function $f(x)$.

Q2: How do we find the composition of two functions?

A2: To find the composition of two functions, we need to substitute the expression for $g(x)$ into the function $f(x)$. This gives us a new function, which is the composition of the two original functions.

Q3: What is the difference between function composition and function addition?

A3: Function composition and function addition are two different mathematical operations. Function composition involves plugging one function into another, while function addition involves adding two functions together.

Q4: Can we find the composition of two functions if they are not defined for the same domain?

A4: Yes, we can find the composition of two functions even if they are not defined for the same domain. However, we need to be careful when plugging in the expression for $g(x)$ into the function $f(x)$, as this may result in a function that is not defined for the entire domain.

Q5: How do we evaluate the composition of two functions at a specific point?

A5: To evaluate the composition of two functions at a specific point, we need to plug in the value of the point into the expression for the composition. This will give us the value of the composition at that point.

Q6: Can we use function composition to solve real-world problems?

A6: Yes, function composition can be used to solve real-world problems. For example, we can use function composition to model population growth, financial transactions, and other complex systems.

Q7: What are some common applications of function composition?

A7: Some common applications of function composition include:

• Modeling population growth
• Financial transactions
• Signal processing
• Image processing
• Data analysis

Q8: Can we use function composition to simplify complex functions?

A8: Yes, function composition can be used to simplify complex functions. By breaking down a complex function into smaller, more manageable pieces, we can use function composition to simplify the function and make it easier to work with.

Q9: How do we know if a function is composite or not?

A9: A function is composite if it can be expressed as the composition of two or more other functions. To determine if a function is composite, we need to look for patterns or structures in the function that suggest it can be broken down into smaller pieces.

Q10: Can we use function composition to create new functions?

A10: Yes, function composition can be used to create new functions. By combining two or more functions in a specific way, we can create a new function that has properties and behaviors that are different from the original functions.

Conclusion

In this article, we answered some frequently asked questions related to the composition of functions and provided additional insights. We hope that this article has been helpful in clarifying the concept of function composition and its applications.

Final Answer

The final answer is $\boxed{63}$.