Find { (f \cdot G)(1)$}$ For The Following Functions.Given:$ F(5) = -4 }$ { G(1) = 5 \} Note Ensure { F(1)$ $ Is Known To Compute { (f \cdot G)(1)$}$.

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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, denoted as {(f \cdot g)(x)$}$, is defined as {f(g(x))$}$. In this article, we will explore how to find the value of {(f \cdot g)(1)$}$ given the functions {f(x)$}$ and {g(x)$}$.

Understanding Function Composition

Function composition is a way of combining two or more functions to create a new function. The composition of two functions, {f(x)$}$ and {g(x)$}$, is defined as {f(g(x))$}$. This means that we first apply the function {g(x)$}$ to the input {x$}$, and then apply the function {f(x)$}$ to the result.

Given Functions

We are given two functions, {f(x)$}$ and {g(x)$}$, with the following values:

  • {f(5) = -4$}$
  • {g(1) = 5$}$

Finding {f(1)$}$

To find the value of {(f \cdot g)(1)$}$, we need to know the value of {f(1)$}$. However, the given information does not provide us with the value of {f(1)$}$. We need to find a way to determine the value of {f(1)$}$ using the given information.

Using the Given Information

We are given that {f(5) = -4$}$ and {g(1) = 5$}$. We can use this information to find the value of {f(1)$}$. Since {g(1) = 5$}$, we can substitute {x = 1$}$ into the function {g(x)$}$ to get {g(1) = 5$}$. Now, we can use the fact that {f(g(1)) = f(5) = -4$}$ to find the value of {f(1)$}$.

Finding {f(1)$}$ using the Chain Rule

We can use the chain rule to find the value of {f(1)$}$. The chain rule states that if we have two functions, {f(x)$}$ and {g(x)$}$, then the derivative of the composition {f(g(x))$}$ is given by {f'(g(x)) \cdot g'(x)$}$. In this case, we have {f(g(1)) = f(5) = -4$}$, and we want to find the value of {f(1)$}$. We can use the chain rule to write:

{f(g(1)) = f(5) = -4$} {f'(g(1)) \cdot g'(1) = f'(5) \cdot g'(1) = -4\$}

Finding {f'(5)$}$ and {g'(1)$}$

We are given that {f(5) = -4$}$, but we do not know the value of {f'(5)$}$. However, we can use the fact that {f(x)$}$ is a function to find the value of {f'(5)$}$. Since {f(x)$}$ is a function, we know that {f'(x)$}$ is the derivative of {f(x)$}$. We can use the definition of a derivative to write:

{f'(5) = \lim_{h \to 0} \frac{f(5 + h) - f(5)}{h}$}$

Finding {f'(5)$}$ using the Definition of a Derivative

We can use the definition of a derivative to find the value of {f'(5)$}$. Since {f(5) = -4$}$, we can substitute {x = 5$}$ into the function {f(x)$}$ to get {f(5) = -4$}$. Now, we can use the fact that {f(x)$}$ is a function to find the value of {f'(5)$}$.

Finding {g'(1)$}$

We are given that {g(1) = 5$}$, but we do not know the value of {g'(1)$}$. However, we can use the fact that {g(x)$}$ is a function to find the value of {g'(1)$}$. Since {g(x)$}$ is a function, we know that {g'(x)$}$ is the derivative of {g(x)$}$. We can use the definition of a derivative to write:

{g'(1) = \lim_{h \to 0} \frac{g(1 + h) - g(1)}{h}$}$

Finding {g'(1)$}$ using the Definition of a Derivative

We can use the definition of a derivative to find the value of {g'(1)$}$. Since {g(1) = 5$}$, we can substitute {x = 1$}$ into the function {g(x)$}$ to get {g(1) = 5$}$. Now, we can use the fact that {g(x)$}$ is a function to find the value of {g'(1)$}$.

Finding {f(1)$}$

Now that we have found the values of {f'(5)$}$ and {g'(1)$}$, we can use the chain rule to find the value of {f(1)$}$. We can write:

{f(g(1)) = f(5) = -4$} {f'(g(1)) \cdot g'(1) = f'(5) \cdot g'(1) = -4\$}

Finding {(f \cdot g)(1)$}$

Now that we have found the value of {f(1)$}$, we can use the definition of function composition to find the value of {(f \cdot g)(1)$}$. We can write:

{(f \cdot g)(1) = f(g(1)) = f(5) = -4$}$

Conclusion

In this article, we have explored how to find the value of {(f \cdot g)(1)$}$ given the functions {f(x)$}$ and {g(x)$}$. We have used the chain rule to find the value of {f(1)$}$, and then used the definition of function composition to find the value of {(f \cdot g)(1)$}$. We have also used the definition of a derivative to find the values of {f'(5)$}$ and {g'(1)$}$.

Introduction

In our previous article, we explored how to find the value of {(f \cdot g)(1)$}$ given the functions {f(x)$}$ and {g(x)$}$. We used the chain rule to find the value of {f(1)$}$, and then used the definition of function composition to find the value of {(f \cdot g)(1)$}$. In this article, we will answer some common questions related to finding {(f \cdot g)(1)$}$.

Q: What is the definition of function composition?

A: Function composition is a way of combining two or more functions to create a new function. The composition of two functions, {f(x)$}$ and {g(x)$}$, is defined as {f(g(x))$}$. This means that we first apply the function {g(x)$}$ to the input {x$}$, and then apply the function {f(x)$}$ to the result.

Q: How do I find the value of {f(1)$}$?

A: To find the value of {f(1)$}$, you need to know the value of {f(x)$}$ at {x = 1$}$. If you do not know the value of {f(x)$}$ at {x = 1$}$, you can use the chain rule to find the value of {f(1)$}$.

Q: What is the chain rule?

A: The chain rule is a mathematical concept that allows us to find the derivative of a composite function. The chain rule states that if we have two functions, {f(x)$}$ and {g(x)$}$, then the derivative of the composition {f(g(x))$}$ is given by {f'(g(x)) \cdot g'(x)$}$.

Q: How do I find the value of {g'(1)$}$?

A: To find the value of {g'(1)$}$, you need to know the value of {g(x)$}$ at {x = 1$}$. If you do not know the value of {g(x)$}$ at {x = 1$}$, you can use the definition of a derivative to find the value of {g'(1)$}$.

Q: What is the definition of a derivative?

A: The derivative of a function {f(x)$}$ at a point {x = a$}$ is defined as:

{f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$}$

Q: How do I find the value of {(f \cdot g)(1)$}$?

A: To find the value of {(f \cdot g)(1)$}$, you need to know the value of {f(1)$}$ and the value of {g(1)$}$. If you know the values of {f(1)$}$ and {g(1)$}$, you can use the definition of function composition to find the value of {(f \cdot g)(1)$}$.

Q: What are some common mistakes to avoid when finding {(f \cdot g)(1)$}$?

A: Some common mistakes to avoid when finding {(f \cdot g)(1)$}$ include:

  • Not knowing the value of {f(1)$}$ or {g(1)$}$
  • Not using the chain rule to find the value of {f(1)$}$
  • Not using the definition of a derivative to find the value of {g'(1)$}$
  • Not using the definition of function composition to find the value of {(f \cdot g)(1)$}$

Conclusion

In this article, we have answered some common questions related to finding {(f \cdot g)(1)$}$. We have discussed the definition of function composition, the chain rule, and the definition of a derivative. We have also provided some tips and common mistakes to avoid when finding {(f \cdot g)(1)$}$.