Given The Functions: F ( X ) = − 2 X F(x) = -2x F ( X ) = − 2 X G ( X ) = ∣ X − 7 ∣ G(x) = |x-7| G ( X ) = ∣ X − 7∣ H ( X ) = 1 X − 3 H(x) = \frac{1}{x-3} H ( X ) = X − 3 1 ​ Evaluate The Function ( F ⋅ G ) ( − 4 (f \cdot G)(-4 ( F ⋅ G ) ( − 4 ] For The Given Value Of X X X . Write Your Answer As An Integer Or Simplified Fraction. Select

Introduction

In mathematics, composite functions are a fundamental concept that allows us to combine multiple functions to create a new function. In this article, we will explore how to evaluate composite functions, with a focus on the given functions $f(x) = -2x$, $g(x) = |x-7|$, and $h(x) = \frac{1}{x-3}$. We will specifically evaluate the function $(f \cdot g)(-4)$ for the given value of $x$.

Understanding Composite Functions

A composite function is a function that is derived from two or more functions. It is denoted by the symbol $\circ$ or $\cdot$. For example, if we have two functions $f(x)$ and $g(x)$, the composite function $(f \cdot g)(x)$ is defined as:

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

In other words, we first apply the function $g(x)$ to the input $x$, and then apply the function $f(x)$ to the result.

Evaluating the Function $(f \cdot g)(-4)$

To evaluate the function $(f \cdot g)(-4)$, we need to follow the order of operations. First, we need to evaluate the function $g(-4)$.

Evaluating the Function $g(-4)$

The function $g(x) = |x-7|$ is an absolute value function. To evaluate $g(-4)$, we need to substitute $x = -4$ into the function:

$$g(-4) = |-4-7| = |-11| = 11$$

Evaluating the Function $f(g(-4))$

Now that we have evaluated the function $g(-4)$, we can substitute the result into the function $f(x) = -2x$:

$$f(g(-4)) = f(11) = -2(11) = -22$$

Therefore, the value of the function $(f \cdot g)(-4)$ is $-22$.

Conclusion

In this article, we have evaluated the composite function $(f \cdot g)(-4)$ for the given value of $x$. We have followed the order of operations and used the definitions of the functions $f(x)$ and $g(x)$ to arrive at the final result. This example illustrates the importance of understanding composite functions and how to evaluate them.

Additional Examples

To further illustrate the concept of composite functions, let's consider a few more examples.

Evaluating the Function $(h \cdot f)(x)$

The function $(h \cdot f)(x)$ is defined as:

$$(h \cdot f)(x) = h(f(x))$$

To evaluate this function, we need to substitute the function $f(x) = -2x$ into the function $h(x) = \frac{1}{x-3}$:

$$(h \cdot f)(x) = h(f(x)) = h(-2x) = \frac{1}{-2x-3}$$

Evaluating the Function $(g \cdot h)(x)$

The function $(g \cdot h)(x)$ is defined as:

$$(g \cdot h)(x) = g(h(x))$$

To evaluate this function, we need to substitute the function $h(x) = \frac{1}{x-3}$ into the function $g(x) = |x-7|$:

$$(g \cdot h)(x) = g(h(x)) = g\left(\frac{1}{x-3}\right) = \left|\frac{1}{x-3}-7\right|$$

These examples demonstrate how to evaluate composite functions and how to use the definitions of the individual functions to arrive at the final result.

Final Thoughts

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Introduction

In our previous article, we explored the concept of composite functions and how to evaluate them. In this article, we will provide a Q&A guide to help you better understand composite functions and how to evaluate them.

Q: What is a composite function?

A: A composite function is a function that is derived from two or more functions. It is denoted by the symbol $\circ$ or $\cdot$. For example, if we have two functions $f(x)$ and $g(x)$, the composite function $(f \cdot g)(x)$ is defined as:

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

Q: How do I evaluate a composite function?

A: To evaluate a composite function, you need to follow the order of operations. First, you need to evaluate the inner function, and then you need to substitute the result into the outer function.

Q: What is the difference between $(f \cdot g)(x)$ and $f(g(x))$?

A: The symbols $(f \cdot g)(x)$ and $f(g(x))$ are used to denote the same composite function. The symbol $\cdot$ is often used to denote function composition, while the parentheses are used to indicate the order of operations.

Q: Can I evaluate a composite function with multiple functions?

A: Yes, you can evaluate a composite function with multiple functions. For example, if we have three functions $f(x)$, $g(x)$, and $h(x)$, we can evaluate the composite function $(f \cdot g \cdot h)(x)$ as follows:

$$(f \cdot g \cdot h)(x) = f(g(h(x)))$$

Q: How do I know which function to evaluate first?

A: To determine which function to evaluate first, you need to look at the order of operations. The innermost function is evaluated first, and then the result is substituted into the next function.

Q: Can I use absolute value functions in composite functions?

A: Yes, you can use absolute value functions in composite functions. For example, if we have the function $g(x) = |x-7|$, we can evaluate the composite function $(f \cdot g)(x)$ as follows:

$$(f \cdot g)(x) = f(g(x)) = f(|x-7|)$$

Q: Can I use rational functions in composite functions?

A: Yes, you can use rational functions in composite functions. For example, if we have the function $h(x) = \frac{1}{x-3}$, we can evaluate the composite function $(f \cdot h)(x)$ as follows:

$$(f \cdot h)(x) = f(h(x)) = f\left(\frac{1}{x-3}\right)$$

Q: How do I simplify a composite function?

A: To simplify a composite function, you need to look for any common factors or terms that can be canceled out. You can also use algebraic manipulations to simplify the function.

Q: Can I use composite functions in real-world applications?

A: Yes, composite functions are used in many real-world applications, such as physics, engineering, and economics. They are used to model complex systems and relationships between variables.

Conclusion

In this Q&A guide, we have provided answers to common questions about composite functions and how to evaluate them. We hope that this guide has been helpful in illustrating the concept of composite functions and how to evaluate them.